[{"data":1,"prerenderedAt":12727},["ShallowReactive",2],{"post-\u002Fblog\u002F2024\u002F2024-09-11-linear-congruential-for-psedurandom-number":3},{"id":4,"title":5,"body":6,"description":11944,"draft":12714,"enableComment":12037,"extension":12715,"image":12716,"meta":12717,"navigation":12037,"onday":12718,"path":12719,"seo":12720,"stem":12721,"summary":12722,"tags":12723,"__hash__":12726},"blog\u002Fblog\u002F2024\u002F2024-09-11-linear-congruential-for-psedurandom-number.md","Linear Congruential for Pseudorandom Number",{"type":7,"value":8,"toc":12704},"minimark",[9,14,18,32,35,38,41,323,414,562,1050,1125,1129,1136,2295,3766,4374,4406,4410,4413,4683,4686,4906,4967,4970,5196,5199,5406,5409,5602,5756,5885,6247,6250,8484,8557,8732,8735,9461,10018,10021,10153,10990,11887,11890,11894,11897,11923,11926,11930,11933,11938,12392,12396,12399,12697,12700],[10,11,13],"h2",{"id":12},"algorithm-overview","Algorithm Overview",[15,16,17],"p",{},"Programmers are generally familiar with random numbers, and it is well-known that the random numbers generated by computers are often pseudo-random. What exactly is a pseudo-random sequence?",[19,20,21],"blockquote",{},[15,22,23,27,28,31],{},[24,25,26],"strong",{},"Pseudo-random numbers"," are a sequence of numbers that appear random but are generated using a ",[24,29,30],{},"deterministic"," algorithm, meaning they are not truly random.",[15,33,34],{},"Since randomness is simulated through an algorithm, what kind of algorithm can achieve an effect close to true randomness, and how does it work?",[15,36,37],{},"One of the simpler methods is the linear congruential generator (LCG).",[15,39,40],{},"The LCG is based on the recursive formula:",[42,43,47],"span",{"className":44,"translate":46},[45],"katex-display","no",[42,48,51,125],{"className":49,"translate":46},[50],"katex",[42,52,55],{"className":53},[54],"katex-mathml",[56,57,60],"math",{"xmlns":58,"display":59},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML","block",[61,62,63,120],"semantics",{},[64,65,66,86,89,93,96,99,105,107,110,113,117],"mrow",{},[67,68,69,73],"msub",{},[70,71,72],"mi",{},"N",[64,74,75,78,82],{},[70,76,77],{},"j",[79,80,81],"mo",{},"+",[83,84,85],"mn",{},"1",[79,87,88],{},"=",[79,90,92],{"stretchy":91},"false","(",[70,94,95],{},"A",[79,97,98],{},"∗",[67,100,101,103],{},[70,102,72],{},[70,104,77],{},[79,106,81],{},[70,108,109],{},"B",[79,111,112],{"stretchy":91},")",[70,114,116],{"mathvariant":115},"normal","%",[70,118,119],{},"M",[121,122,124],"annotation",{"encoding":123},"application\u002Fx-tex","N_{j+1} = (A * N_{j} + B) \\% M",[42,126,130,221,245,303],{"className":127,"ariaHidden":129},[128],"katex-html","true",[42,131,134,139,209,214,218],{"className":132},[133],"base",[42,135],{"className":136,"style":138},[137],"strut","height:0.9694em;vertical-align:-0.2861em;",[42,140,143,148],{"className":141},[142],"mord",[42,144,72],{"className":145,"style":147},[142,146],"mathnormal","margin-right:0.10903em;",[42,149,152],{"className":150},[151],"msupsub",[42,153,157,200],{"className":154},[155,156],"vlist-t","vlist-t2",[42,158,161,195],{"className":159},[160],"vlist-r",[42,162,166],{"className":163,"style":165},[164],"vlist","height:0.3117em;",[42,167,169,174],{"style":168},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[42,170],{"className":171,"style":173},[172],"pstrut","height:2.7em;",[42,175,181],{"className":176},[177,178,179,180],"sizing","reset-size6","size3","mtight",[42,182,184,188,192],{"className":183},[142,180],[42,185,77],{"className":186,"style":187},[142,146,180],"margin-right:0.05724em;",[42,189,81],{"className":190},[191,180],"mbin",[42,193,85],{"className":194},[142,180],[42,196,199],{"className":197},[198],"vlist-s","​",[42,201,203],{"className":202},[160],[42,204,207],{"className":205,"style":206},[164],"height:0.2861em;",[42,208],{},[42,210],{"className":211,"style":213},[212],"mspace","margin-right:0.2778em;",[42,215,88],{"className":216},[217],"mrel",[42,219],{"className":220,"style":213},[212],[42,222,224,228,232,235,239,242],{"className":223},[133],[42,225],{"className":226,"style":227},[137],"height:1em;vertical-align:-0.25em;",[42,229,92],{"className":230},[231],"mopen",[42,233,95],{"className":234},[142,146],[42,236],{"className":237,"style":238},[212],"margin-right:0.2222em;",[42,240,98],{"className":241},[191],[42,243],{"className":244,"style":238},[212],[42,246,248,251,294,297,300],{"className":247},[133],[42,249],{"className":250,"style":138},[137],[42,252,254,257],{"className":253},[142],[42,255,72],{"className":256,"style":147},[142,146],[42,258,260],{"className":259},[151],[42,261,263,286],{"className":262},[155,156],[42,264,266,283],{"className":265},[160],[42,267,269],{"className":268,"style":165},[164],[42,270,271,274],{"style":168},[42,272],{"className":273,"style":173},[172],[42,275,277],{"className":276},[177,178,179,180],[42,278,280],{"className":279},[142,180],[42,281,77],{"className":282,"style":187},[142,146,180],[42,284,199],{"className":285},[198],[42,287,289],{"className":288},[160],[42,290,292],{"className":291,"style":206},[164],[42,293],{},[42,295],{"className":296,"style":238},[212],[42,298,81],{"className":299},[191],[42,301],{"className":302,"style":238},[212],[42,304,306,309,313,317,320],{"className":305},[133],[42,307],{"className":308,"style":227},[137],[42,310,109],{"className":311,"style":312},[142,146],"margin-right:0.05017em;",[42,314,112],{"className":315},[316],"mclose",[42,318,116],{"className":319},[142],[42,321,119],{"className":322,"style":147},[142,146],[15,324,325,326,355,356,384,385,413],{},"The most important elements in the LCG are three integers: the multiplier ",[42,327,329,342],{"className":328,"translate":46},[50],[42,330,332],{"className":331},[54],[56,333,334],{"xmlns":58},[61,335,336,340],{},[64,337,338],{},[70,339,95],{},[121,341,95],{"encoding":123},[42,343,345],{"className":344,"ariaHidden":129},[128],[42,346,348,352],{"className":347},[133],[42,349],{"className":350,"style":351},[137],"height:0.6833em;",[42,353,95],{"className":354},[142,146],", the increment ",[42,357,359,372],{"className":358,"translate":46},[50],[42,360,362],{"className":361},[54],[56,363,364],{"xmlns":58},[61,365,366,370],{},[64,367,368],{},[70,369,109],{},[121,371,109],{"encoding":123},[42,373,375],{"className":374,"ariaHidden":129},[128],[42,376,378,381],{"className":377},[133],[42,379],{"className":380,"style":351},[137],[42,382,109],{"className":383,"style":312},[142,146],", and the modulus ",[42,386,388,401],{"className":387,"translate":46},[50],[42,389,391],{"className":390},[54],[56,392,393],{"xmlns":58},[61,394,395,399],{},[64,396,397],{},[70,398,119],{},[121,400,119],{"encoding":123},[42,402,404],{"className":403,"ariaHidden":129},[128],[42,405,407,410],{"className":406},[133],[42,408],{"className":409,"style":351},[137],[42,411,119],{"className":412,"style":147},[142,146],".",[15,415,416,417,445,446,474,475,503,504,532,533,561],{},"These values, ",[42,418,420,433],{"className":419,"translate":46},[50],[42,421,423],{"className":422},[54],[56,424,425],{"xmlns":58},[61,426,427,431],{},[64,428,429],{},[70,430,95],{},[121,432,95],{"encoding":123},[42,434,436],{"className":435,"ariaHidden":129},[128],[42,437,439,442],{"className":438},[133],[42,440],{"className":441,"style":351},[137],[42,443,95],{"className":444},[142,146],", ",[42,447,449,462],{"className":448,"translate":46},[50],[42,450,452],{"className":451},[54],[56,453,454],{"xmlns":58},[61,455,456,460],{},[64,457,458],{},[70,459,109],{},[121,461,109],{"encoding":123},[42,463,465],{"className":464,"ariaHidden":129},[128],[42,466,468,471],{"className":467},[133],[42,469],{"className":470,"style":351},[137],[42,472,109],{"className":473,"style":312},[142,146],", and ",[42,476,478,491],{"className":477,"translate":46},[50],[42,479,481],{"className":480},[54],[56,482,483],{"xmlns":58},[61,484,485,489],{},[64,486,487],{},[70,488,119],{},[121,490,119],{"encoding":123},[42,492,494],{"className":493,"ariaHidden":129},[128],[42,495,497,500],{"className":496},[133],[42,498],{"className":499,"style":351},[137],[42,501,119],{"className":502,"style":147},[142,146],", are constants set by the generator. The period of the LCG is at most ",[42,505,507,520],{"className":506,"translate":46},[50],[42,508,510],{"className":509},[54],[56,511,512],{"xmlns":58},[61,513,514,518],{},[64,515,516],{},[70,517,119],{},[121,519,119],{"encoding":123},[42,521,523],{"className":522,"ariaHidden":129},[128],[42,524,526,529],{"className":525},[133],[42,527],{"className":528,"style":351},[137],[42,530,119],{"className":531,"style":147},[142,146],", though in most cases, it will be less than ",[42,534,536,549],{"className":535,"translate":46},[50],[42,537,539],{"className":538},[54],[56,540,541],{"xmlns":58},[61,542,543,547],{},[64,544,545],{},[70,546,119],{},[121,548,119],{"encoding":123},[42,550,552],{"className":551,"ariaHidden":129},[128],[42,553,555,558],{"className":554},[133],[42,556],{"className":557,"style":351},[137],[42,559,119],{"className":560,"style":147},[142,146],". To achieve the maximum period, the following conditions must be met:",[563,564,565,626,713,830],"ul",{},[566,567,568,596,597,625],"li",{},[42,569,571,584],{"className":570,"translate":46},[50],[42,572,574],{"className":573},[54],[56,575,576],{"xmlns":58},[61,577,578,582],{},[64,579,580],{},[70,581,109],{},[121,583,109],{"encoding":123},[42,585,587],{"className":586,"ariaHidden":129},[128],[42,588,590,593],{"className":589},[133],[42,591],{"className":592,"style":351},[137],[42,594,109],{"className":595,"style":312},[142,146]," and ",[42,598,600,613],{"className":599,"translate":46},[50],[42,601,603],{"className":602},[54],[56,604,605],{"xmlns":58},[61,606,607,611],{},[64,608,609],{},[70,610,119],{},[121,612,119],{"encoding":123},[42,614,616],{"className":615,"ariaHidden":129},[128],[42,617,619,622],{"className":618},[133],[42,620],{"className":621,"style":351},[137],[42,623,119],{"className":624,"style":147},[142,146]," are coprime;",[566,627,628,629,657,658,712],{},"All prime factors of ",[42,630,632,645],{"className":631,"translate":46},[50],[42,633,635],{"className":634},[54],[56,636,637],{"xmlns":58},[61,638,639,643],{},[64,640,641],{},[70,642,119],{},[121,644,119],{"encoding":123},[42,646,648],{"className":647,"ariaHidden":129},[128],[42,649,651,654],{"className":650},[133],[42,652],{"className":653,"style":351},[137],[42,655,119],{"className":656,"style":147},[142,146]," divide ",[42,659,661,680],{"className":660,"translate":46},[50],[42,662,664],{"className":663},[54],[56,665,666],{"xmlns":58},[61,667,668,677],{},[64,669,670,672,675],{},[70,671,95],{},[79,673,674],{},"−",[83,676,85],{},[121,678,679],{"encoding":123},"A-1",[42,681,683,702],{"className":682,"ariaHidden":129},[128],[42,684,686,690,693,696,699],{"className":685},[133],[42,687],{"className":688,"style":689},[137],"height:0.7667em;vertical-align:-0.0833em;",[42,691,95],{"className":692},[142,146],[42,694],{"className":695,"style":238},[212],[42,697,674],{"className":698},[191],[42,700],{"className":701,"style":238},[212],[42,703,705,709],{"className":704},[133],[42,706],{"className":707,"style":708},[137],"height:0.6444em;",[42,710,85],{"className":711},[142],";",[566,714,715,716,778,779,829],{},"If ",[42,717,719,743],{"className":718,"translate":46},[50],[42,720,722],{"className":721},[54],[56,723,724],{"xmlns":58},[61,725,726,740],{},[64,727,728,730,732,735,737],{},[70,729,119],{},[70,731,116],{"mathvariant":115},[83,733,734],{},"4",[79,736,88],{},[83,738,739],{},"0",[121,741,742],{"encoding":123},"M \\% 4 = 0",[42,744,746,769],{"className":745,"ariaHidden":129},[128],[42,747,749,753,756,760,763,766],{"className":748},[133],[42,750],{"className":751,"style":752},[137],"height:0.8056em;vertical-align:-0.0556em;",[42,754,119],{"className":755,"style":147},[142,146],[42,757,759],{"className":758},[142],"%4",[42,761],{"className":762,"style":213},[212],[42,764,88],{"className":765},[217],[42,767],{"className":768,"style":213},[212],[42,770,772,775],{"className":771},[133],[42,773],{"className":774,"style":708},[137],[42,776,739],{"className":777},[142],", then ",[42,780,782,799],{"className":781,"translate":46},[50],[42,783,785],{"className":784},[54],[56,786,787],{"xmlns":58},[61,788,789,797],{},[64,790,791,793,795],{},[70,792,95],{},[79,794,674],{},[83,796,85],{},[121,798,679],{"encoding":123},[42,800,802,820],{"className":801,"ariaHidden":129},[128],[42,803,805,808,811,814,817],{"className":804},[133],[42,806],{"className":807,"style":689},[137],[42,809,95],{"className":810},[142,146],[42,812],{"className":813,"style":238},[212],[42,815,674],{"className":816},[191],[42,818],{"className":819,"style":238},[212],[42,821,823,826],{"className":822},[133],[42,824],{"className":825,"style":708},[137],[42,827,85],{"className":828},[142]," must also be divisible by 4;",[566,831,832,445,860,474,888,964,965,474,993,445,1021,1049],{},[42,833,835,848],{"className":834,"translate":46},[50],[42,836,838],{"className":837},[54],[56,839,840],{"xmlns":58},[61,841,842,846],{},[64,843,844],{},[70,845,95],{},[121,847,95],{"encoding":123},[42,849,851],{"className":850,"ariaHidden":129},[128],[42,852,854,857],{"className":853},[133],[42,855],{"className":856,"style":351},[137],[42,858,95],{"className":859},[142,146],[42,861,863,876],{"className":862,"translate":46},[50],[42,864,866],{"className":865},[54],[56,867,868],{"xmlns":58},[61,869,870,874],{},[64,871,872],{},[70,873,109],{},[121,875,109],{"encoding":123},[42,877,879],{"className":878,"ariaHidden":129},[128],[42,880,882,885],{"className":881},[133],[42,883],{"className":884,"style":351},[137],[42,886,109],{"className":887,"style":312},[142,146],[42,889,891,909],{"className":890,"translate":46},[50],[42,892,894],{"className":893},[54],[56,895,896],{"xmlns":58},[61,897,898,906],{},[64,899,900],{},[67,901,902,904],{},[70,903,72],{},[83,905,739],{},[121,907,908],{"encoding":123},"N_{0}",[42,910,912],{"className":911,"ariaHidden":129},[128],[42,913,915,919],{"className":914},[133],[42,916],{"className":917,"style":918},[137],"height:0.8333em;vertical-align:-0.15em;",[42,920,922,925],{"className":921},[142],[42,923,72],{"className":924,"style":147},[142,146],[42,926,928],{"className":927},[151],[42,929,931,955],{"className":930},[155,156],[42,932,934,952],{"className":933},[160],[42,935,938],{"className":936,"style":937},[164],"height:0.3011em;",[42,939,940,943],{"style":168},[42,941],{"className":942,"style":173},[172],[42,944,946],{"className":945},[177,178,179,180],[42,947,949],{"className":948},[142,180],[42,950,739],{"className":951},[142,180],[42,953,199],{"className":954},[198],[42,956,958],{"className":957},[160],[42,959,962],{"className":960,"style":961},[164],"height:0.15em;",[42,963],{}," must be smaller than ",[42,966,968,981],{"className":967,"translate":46},[50],[42,969,971],{"className":970},[54],[56,972,973],{"xmlns":58},[61,974,975,979],{},[64,976,977],{},[70,978,119],{},[121,980,119],{"encoding":123},[42,982,984],{"className":983,"ariaHidden":129},[128],[42,985,987,990],{"className":986},[133],[42,988],{"className":989,"style":351},[137],[42,991,119],{"className":992,"style":147},[142,146],[42,994,996,1009],{"className":995,"translate":46},[50],[42,997,999],{"className":998},[54],[56,1000,1001],{"xmlns":58},[61,1002,1003,1007],{},[64,1004,1005],{},[70,1006,95],{},[121,1008,95],{"encoding":123},[42,1010,1012],{"className":1011,"ariaHidden":129},[128],[42,1013,1015,1018],{"className":1014},[133],[42,1016],{"className":1017,"style":351},[137],[42,1019,95],{"className":1020},[142,146],[42,1022,1024,1037],{"className":1023,"translate":46},[50],[42,1025,1027],{"className":1026},[54],[56,1028,1029],{"xmlns":58},[61,1030,1031,1035],{},[64,1032,1033],{},[70,1034,109],{},[121,1036,109],{"encoding":123},[42,1038,1040],{"className":1039,"ariaHidden":129},[128],[42,1041,1043,1046],{"className":1042},[133],[42,1044],{"className":1045,"style":351},[137],[42,1047,109],{"className":1048,"style":312},[142,146]," must be positive integers.",[15,1051,1052,1053,1056,1057,1124],{},"The advantage of the LCG is that it is ",[24,1054,1055],{},"very easy to implement and generates numbers quickly",". However, its drawback is also clear: for a 32-bit number, the maximum period is limited to ",[42,1058,1060,1081],{"className":1059,"translate":46},[50],[42,1061,1063],{"className":1062},[54],[56,1064,1065],{"xmlns":58},[61,1066,1067,1078],{},[64,1068,1069],{},[1070,1071,1072,1075],"msup",{},[83,1073,1074],{},"2",[83,1076,1077],{},"32",[121,1079,1080],{"encoding":123},"2^{32}",[42,1082,1084],{"className":1083,"ariaHidden":129},[128],[42,1085,1087,1091],{"className":1086},[133],[42,1088],{"className":1089,"style":1090},[137],"height:0.8141em;",[42,1092,1094,1097],{"className":1093},[142],[42,1095,1074],{"className":1096},[142],[42,1098,1100],{"className":1099},[151],[42,1101,1103],{"className":1102},[155],[42,1104,1106],{"className":1105},[160],[42,1107,1109],{"className":1108,"style":1090},[164],[42,1110,1112,1115],{"style":1111},"top:-3.063em;margin-right:0.05em;",[42,1113],{"className":1114,"style":173},[172],[42,1116,1118],{"className":1117},[177,178,179,180],[42,1119,1121],{"className":1120},[142,180],[42,1122,1077],{"className":1123},[142,180],", which is inadequate for applications that require high-quality random numbers, such as cryptography.",[10,1126,1128],{"id":1127},"randomness-analysis","Randomness Analysis",[15,1130,1131,1132,1135],{},"With this understanding, one might wonder, how does the LCG simulate a random process? True randomness implies that any number can appear, meaning ",[24,1133,1134],{},"all outcomes are equally probable, following a uniform distribution",". So, how does the LCG achieve uniform distribution?",[15,1137,1138,1139,1167,1168,1196,1197,1462,1463,778,1493,1649,1650,1903,1904,1977,1978,2063,2064,2092,2093,2121,2122,2150,2151,2179,2180,2241,2242,413],{},"First, we need to understand the period of the LCG. From the definition above, it is clear that for a modulus ",[42,1140,1142,1155],{"className":1141,"translate":46},[50],[42,1143,1145],{"className":1144},[54],[56,1146,1147],{"xmlns":58},[61,1148,1149,1153],{},[64,1150,1151],{},[70,1152,119],{},[121,1154,119],{"encoding":123},[42,1156,1158],{"className":1157,"ariaHidden":129},[128],[42,1159,1161,1164],{"className":1160},[133],[42,1162],{"className":1163,"style":351},[137],[42,1165,119],{"className":1166,"style":147},[142,146],", the sequence generated by the LCG has a maximum period of ",[42,1169,1171,1184],{"className":1170,"translate":46},[50],[42,1172,1174],{"className":1173},[54],[56,1175,1176],{"xmlns":58},[61,1177,1178,1182],{},[64,1179,1180],{},[70,1181,119],{},[121,1183,119],{"encoding":123},[42,1185,1187],{"className":1186,"ariaHidden":129},[128],[42,1188,1190,1193],{"className":1189},[133],[42,1191],{"className":1192,"style":351},[137],[42,1194,119],{"className":1195,"style":147},[142,146],". Suppose the recursive formula is ",[42,1198,1200,1268],{"className":1199,"translate":46},[50],[42,1201,1203],{"className":1202},[54],[56,1204,1205],{"xmlns":58},[61,1206,1207,1265],{},[64,1208,1209,1221,1223,1225,1227,1229,1235,1237,1239,1241,1243,1246,1257,1261,1263],{},[67,1210,1211,1213],{},[70,1212,72],{},[64,1214,1215,1217,1219],{},[70,1216,77],{},[79,1218,81],{},[83,1220,85],{},[79,1222,88],{},[79,1224,92],{"stretchy":91},[70,1226,95],{},[79,1228,98],{},[67,1230,1231,1233],{},[70,1232,72],{},[70,1234,77],{},[79,1236,81],{},[70,1238,109],{},[79,1240,112],{"stretchy":91},[212,1242],{},[212,1244],{"width":1245},"0.6667em",[64,1247,1248,1251,1254],{},[70,1249,1250],{"mathvariant":115},"m",[70,1252,1253],{"mathvariant":115},"o",[70,1255,1256],{"mathvariant":115},"d",[1258,1259,1260],"mtext",{}," ",[1258,1262,1260],{},[70,1264,119],{},[121,1266,1267],{"encoding":123},"N_{j+1} = (A * N_{j} + B) \\mod M",[42,1269,1271,1335,1356,1414,1434],{"className":1270,"ariaHidden":129},[128],[42,1272,1274,1277,1326,1329,1332],{"className":1273},[133],[42,1275],{"className":1276,"style":138},[137],[42,1278,1280,1283],{"className":1279},[142],[42,1281,72],{"className":1282,"style":147},[142,146],[42,1284,1286],{"className":1285},[151],[42,1287,1289,1318],{"className":1288},[155,156],[42,1290,1292,1315],{"className":1291},[160],[42,1293,1295],{"className":1294,"style":165},[164],[42,1296,1297,1300],{"style":168},[42,1298],{"className":1299,"style":173},[172],[42,1301,1303],{"className":1302},[177,178,179,180],[42,1304,1306,1309,1312],{"className":1305},[142,180],[42,1307,77],{"className":1308,"style":187},[142,146,180],[42,1310,81],{"className":1311},[191,180],[42,1313,85],{"className":1314},[142,180],[42,1316,199],{"className":1317},[198],[42,1319,1321],{"className":1320},[160],[42,1322,1324],{"className":1323,"style":206},[164],[42,1325],{},[42,1327],{"className":1328,"style":213},[212],[42,1330,88],{"className":1331},[217],[42,1333],{"className":1334,"style":213},[212],[42,1336,1338,1341,1344,1347,1350,1353],{"className":1337},[133],[42,1339],{"className":1340,"style":227},[137],[42,1342,92],{"className":1343},[231],[42,1345,95],{"className":1346},[142,146],[42,1348],{"className":1349,"style":238},[212],[42,1351,98],{"className":1352},[191],[42,1354],{"className":1355,"style":238},[212],[42,1357,1359,1362,1405,1408,1411],{"className":1358},[133],[42,1360],{"className":1361,"style":138},[137],[42,1363,1365,1368],{"className":1364},[142],[42,1366,72],{"className":1367,"style":147},[142,146],[42,1369,1371],{"className":1370},[151],[42,1372,1374,1397],{"className":1373},[155,156],[42,1375,1377,1394],{"className":1376},[160],[42,1378,1380],{"className":1379,"style":165},[164],[42,1381,1382,1385],{"style":168},[42,1383],{"className":1384,"style":173},[172],[42,1386,1388],{"className":1387},[177,178,179,180],[42,1389,1391],{"className":1390},[142,180],[42,1392,77],{"className":1393,"style":187},[142,146,180],[42,1395,199],{"className":1396},[198],[42,1398,1400],{"className":1399},[160],[42,1401,1403],{"className":1402,"style":206},[164],[42,1404],{},[42,1406],{"className":1407,"style":238},[212],[42,1409,81],{"className":1410},[191],[42,1412],{"className":1413,"style":238},[212],[42,1415,1417,1420,1423,1426,1430],{"className":1416},[133],[42,1418],{"className":1419,"style":227},[137],[42,1421,109],{"className":1422,"style":312},[142,146],[42,1424,112],{"className":1425},[316],[42,1427],{"className":1428},[212,1429],"allowbreak",[42,1431],{"className":1432,"style":1433},[212],"margin-right:0.6667em;",[42,1435,1437,1441,1452,1456,1459],{"className":1436},[133],[42,1438],{"className":1439,"style":1440},[137],"height:0.6944em;",[42,1442,1444],{"className":1443},[142],[42,1445,1447],{"className":1446},[142],[42,1448,1451],{"className":1449},[142,1450],"mathrm","mod",[42,1453],{"className":1454,"style":1455},[212],"margin-right:0.1667em;",[42,1457],{"className":1458,"style":1455},[212],[42,1460,119],{"className":1461,"style":147},[142,146],". If the period is ",[42,1464,1466,1480],{"className":1465,"translate":46},[50],[42,1467,1469],{"className":1468},[54],[56,1470,1471],{"xmlns":58},[61,1472,1473,1478],{},[64,1474,1475],{},[70,1476,1477],{},"T",[121,1479,1477],{"encoding":123},[42,1481,1483],{"className":1482,"ariaHidden":129},[128],[42,1484,1486,1489],{"className":1485},[133],[42,1487],{"className":1488,"style":351},[137],[42,1490,1477],{"className":1491,"style":1492},[142,146],"margin-right:0.13889em;",[42,1494,1496,1529],{"className":1495,"translate":46},[50],[42,1497,1499],{"className":1498},[54],[56,1500,1501],{"xmlns":58},[61,1502,1503,1526],{},[64,1504,1505,1518,1520],{},[67,1506,1507,1509],{},[70,1508,72],{},[64,1510,1511,1514,1516],{},[70,1512,1513],{},"k",[79,1515,81],{},[70,1517,1477],{},[79,1519,88],{},[67,1521,1522,1524],{},[70,1523,72],{},[70,1525,1513],{},[121,1527,1528],{"encoding":123},"N_{k+T} = N_{k}",[42,1530,1532,1600],{"className":1531,"ariaHidden":129},[128],[42,1533,1535,1539,1591,1594,1597],{"className":1534},[133],[42,1536],{"className":1537,"style":1538},[137],"height:0.8917em;vertical-align:-0.2083em;",[42,1540,1542,1545],{"className":1541},[142],[42,1543,72],{"className":1544,"style":147},[142,146],[42,1546,1548],{"className":1547},[151],[42,1549,1551,1582],{"className":1550},[155,156],[42,1552,1554,1579],{"className":1553},[160],[42,1555,1558],{"className":1556,"style":1557},[164],"height:0.3361em;",[42,1559,1560,1563],{"style":168},[42,1561],{"className":1562,"style":173},[172],[42,1564,1566],{"className":1565},[177,178,179,180],[42,1567,1569,1573,1576],{"className":1568},[142,180],[42,1570,1513],{"className":1571,"style":1572},[142,146,180],"margin-right:0.03148em;",[42,1574,81],{"className":1575},[191,180],[42,1577,1477],{"className":1578,"style":1492},[142,146,180],[42,1580,199],{"className":1581},[198],[42,1583,1585],{"className":1584},[160],[42,1586,1589],{"className":1587,"style":1588},[164],"height:0.2083em;",[42,1590],{},[42,1592],{"className":1593,"style":213},[212],[42,1595,88],{"className":1596},[217],[42,1598],{"className":1599,"style":213},[212],[42,1601,1603,1606],{"className":1602},[133],[42,1604],{"className":1605,"style":918},[137],[42,1607,1609,1612],{"className":1608},[142],[42,1610,72],{"className":1611,"style":147},[142,146],[42,1613,1615],{"className":1614},[151],[42,1616,1618,1641],{"className":1617},[155,156],[42,1619,1621,1638],{"className":1620},[160],[42,1622,1624],{"className":1623,"style":1557},[164],[42,1625,1626,1629],{"style":168},[42,1627],{"className":1628,"style":173},[172],[42,1630,1632],{"className":1631},[177,178,179,180],[42,1633,1635],{"className":1634},[142,180],[42,1636,1513],{"className":1637,"style":1572},[142,146,180],[42,1639,199],{"className":1640},[198],[42,1642,1644],{"className":1643},[160],[42,1645,1647],{"className":1646,"style":961},[164],[42,1648],{},", and since ",[42,1651,1653,1715],{"className":1652,"translate":46},[50],[42,1654,1656],{"className":1655},[54],[56,1657,1658],{"xmlns":58},[61,1659,1660,1712],{},[64,1661,1662,1674,1676,1678,1680,1682,1688,1690,1692,1694,1696,1698,1706,1708,1710],{},[67,1663,1664,1666],{},[70,1665,72],{},[64,1667,1668,1670,1672],{},[70,1669,1513],{},[79,1671,81],{},[83,1673,85],{},[79,1675,88],{},[79,1677,92],{"stretchy":91},[70,1679,95],{},[79,1681,98],{},[67,1683,1684,1686],{},[70,1685,72],{},[70,1687,1513],{},[79,1689,81],{},[70,1691,109],{},[79,1693,112],{"stretchy":91},[212,1695],{},[212,1697],{"width":1245},[64,1699,1700,1702,1704],{},[70,1701,1250],{"mathvariant":115},[70,1703,1253],{"mathvariant":115},[70,1705,1256],{"mathvariant":115},[1258,1707,1260],{},[1258,1709,1260],{},[70,1711,119],{},[121,1713,1714],{"encoding":123},"N_{k+1} = (A * N_{k} + B) \\mod M",[42,1716,1718,1782,1803,1861,1879],{"className":1717,"ariaHidden":129},[128],[42,1719,1721,1724,1773,1776,1779],{"className":1720},[133],[42,1722],{"className":1723,"style":1538},[137],[42,1725,1727,1730],{"className":1726},[142],[42,1728,72],{"className":1729,"style":147},[142,146],[42,1731,1733],{"className":1732},[151],[42,1734,1736,1765],{"className":1735},[155,156],[42,1737,1739,1762],{"className":1738},[160],[42,1740,1742],{"className":1741,"style":1557},[164],[42,1743,1744,1747],{"style":168},[42,1745],{"className":1746,"style":173},[172],[42,1748,1750],{"className":1749},[177,178,179,180],[42,1751,1753,1756,1759],{"className":1752},[142,180],[42,1754,1513],{"className":1755,"style":1572},[142,146,180],[42,1757,81],{"className":1758},[191,180],[42,1760,85],{"className":1761},[142,180],[42,1763,199],{"className":1764},[198],[42,1766,1768],{"className":1767},[160],[42,1769,1771],{"className":1770,"style":1588},[164],[42,1772],{},[42,1774],{"className":1775,"style":213},[212],[42,1777,88],{"className":1778},[217],[42,1780],{"className":1781,"style":213},[212],[42,1783,1785,1788,1791,1794,1797,1800],{"className":1784},[133],[42,1786],{"className":1787,"style":227},[137],[42,1789,92],{"className":1790},[231],[42,1792,95],{"className":1793},[142,146],[42,1795],{"className":1796,"style":238},[212],[42,1798,98],{"className":1799},[191],[42,1801],{"className":1802,"style":238},[212],[42,1804,1806,1809,1852,1855,1858],{"className":1805},[133],[42,1807],{"className":1808,"style":918},[137],[42,1810,1812,1815],{"className":1811},[142],[42,1813,72],{"className":1814,"style":147},[142,146],[42,1816,1818],{"className":1817},[151],[42,1819,1821,1844],{"className":1820},[155,156],[42,1822,1824,1841],{"className":1823},[160],[42,1825,1827],{"className":1826,"style":1557},[164],[42,1828,1829,1832],{"style":168},[42,1830],{"className":1831,"style":173},[172],[42,1833,1835],{"className":1834},[177,178,179,180],[42,1836,1838],{"className":1837},[142,180],[42,1839,1513],{"className":1840,"style":1572},[142,146,180],[42,1842,199],{"className":1843},[198],[42,1845,1847],{"className":1846},[160],[42,1848,1850],{"className":1849,"style":961},[164],[42,1851],{},[42,1853],{"className":1854,"style":238},[212],[42,1856,81],{"className":1857},[191],[42,1859],{"className":1860,"style":238},[212],[42,1862,1864,1867,1870,1873,1876],{"className":1863},[133],[42,1865],{"className":1866,"style":227},[137],[42,1868,109],{"className":1869,"style":312},[142,146],[42,1871,112],{"className":1872},[316],[42,1874],{"className":1875},[212,1429],[42,1877],{"className":1878,"style":1433},[212],[42,1880,1882,1885,1894,1897,1900],{"className":1881},[133],[42,1883],{"className":1884,"style":1440},[137],[42,1886,1888],{"className":1887},[142],[42,1889,1891],{"className":1890},[142],[42,1892,1451],{"className":1893},[142,1450],[42,1895],{"className":1896,"style":1455},[212],[42,1898],{"className":1899,"style":1455},[212],[42,1901,119],{"className":1902,"style":147},[142,146],", this implies that each value of ",[42,1905,1907,1925],{"className":1906,"translate":46},[50],[42,1908,1910],{"className":1909},[54],[56,1911,1912],{"xmlns":58},[61,1913,1914,1922],{},[64,1915,1916],{},[67,1917,1918,1920],{},[70,1919,72],{},[70,1921,1513],{},[121,1923,1924],{"encoding":123},"N_{k}",[42,1926,1928],{"className":1927,"ariaHidden":129},[128],[42,1929,1931,1934],{"className":1930},[133],[42,1932],{"className":1933,"style":918},[137],[42,1935,1937,1940],{"className":1936},[142],[42,1938,72],{"className":1939,"style":147},[142,146],[42,1941,1943],{"className":1942},[151],[42,1944,1946,1969],{"className":1945},[155,156],[42,1947,1949,1966],{"className":1948},[160],[42,1950,1952],{"className":1951,"style":1557},[164],[42,1953,1954,1957],{"style":168},[42,1955],{"className":1956,"style":173},[172],[42,1958,1960],{"className":1959},[177,178,179,180],[42,1961,1963],{"className":1962},[142,180],[42,1964,1513],{"className":1965,"style":1572},[142,146,180],[42,1967,199],{"className":1968},[198],[42,1970,1972],{"className":1971},[160],[42,1973,1975],{"className":1974,"style":961},[164],[42,1976],{}," uniquely determines the value of ",[42,1979,1981,2005],{"className":1980,"translate":46},[50],[42,1982,1984],{"className":1983},[54],[56,1985,1986],{"xmlns":58},[61,1987,1988,2002],{},[64,1989,1990],{},[67,1991,1992,1994],{},[70,1993,72],{},[64,1995,1996,1998,2000],{},[70,1997,1513],{},[79,1999,81],{},[83,2001,85],{},[121,2003,2004],{"encoding":123},"N_{k+1}",[42,2006,2008],{"className":2007,"ariaHidden":129},[128],[42,2009,2011,2014],{"className":2010},[133],[42,2012],{"className":2013,"style":1538},[137],[42,2015,2017,2020],{"className":2016},[142],[42,2018,72],{"className":2019,"style":147},[142,146],[42,2021,2023],{"className":2022},[151],[42,2024,2026,2055],{"className":2025},[155,156],[42,2027,2029,2052],{"className":2028},[160],[42,2030,2032],{"className":2031,"style":1557},[164],[42,2033,2034,2037],{"style":168},[42,2035],{"className":2036,"style":173},[172],[42,2038,2040],{"className":2039},[177,178,179,180],[42,2041,2043,2046,2049],{"className":2042},[142,180],[42,2044,1513],{"className":2045,"style":1572},[142,146,180],[42,2047,81],{"className":2048},[191,180],[42,2050,85],{"className":2051},[142,180],[42,2053,199],{"className":2054},[198],[42,2056,2058],{"className":2057},[160],[42,2059,2061],{"className":2060,"style":1588},[164],[42,2062],{},". Thus, the period ",[42,2065,2067,2080],{"className":2066,"translate":46},[50],[42,2068,2070],{"className":2069},[54],[56,2071,2072],{"xmlns":58},[61,2073,2074,2078],{},[64,2075,2076],{},[70,2077,1477],{},[121,2079,1477],{"encoding":123},[42,2081,2083],{"className":2082,"ariaHidden":129},[128],[42,2084,2086,2089],{"className":2085},[133],[42,2087],{"className":2088,"style":351},[137],[42,2090,1477],{"className":2091,"style":1492},[142,146]," must be less than or equal to ",[42,2094,2096,2109],{"className":2095,"translate":46},[50],[42,2097,2099],{"className":2098},[54],[56,2100,2101],{"xmlns":58},[61,2102,2103,2107],{},[64,2104,2105],{},[70,2106,119],{},[121,2108,119],{"encoding":123},[42,2110,2112],{"className":2111,"ariaHidden":129},[128],[42,2113,2115,2118],{"className":2114},[133],[42,2116],{"className":2117,"style":351},[137],[42,2119,119],{"className":2120,"style":147},[142,146],", because there are only ",[42,2123,2125,2138],{"className":2124,"translate":46},[50],[42,2126,2128],{"className":2127},[54],[56,2129,2130],{"xmlns":58},[61,2131,2132,2136],{},[64,2133,2134],{},[70,2135,119],{},[121,2137,119],{"encoding":123},[42,2139,2141],{"className":2140,"ariaHidden":129},[128],[42,2142,2144,2147],{"className":2143},[133],[42,2145],{"className":2146,"style":351},[137],[42,2148,119],{"className":2149,"style":147},[142,146]," possible distinct integers when taken modulo ",[42,2152,2154,2167],{"className":2153,"translate":46},[50],[42,2155,2157],{"className":2156},[54],[56,2158,2159],{"xmlns":58},[61,2160,2161,2165],{},[64,2162,2163],{},[70,2164,119],{},[121,2166,119],{"encoding":123},[42,2168,2170],{"className":2169,"ariaHidden":129},[128],[42,2171,2173,2176],{"className":2172},[133],[42,2174],{"className":2175,"style":351},[137],[42,2177,119],{"className":2178,"style":147},[142,146],". The ",[42,2181,2183,2205],{"className":2182,"translate":46},[50],[42,2184,2186],{"className":2185},[54],[56,2187,2188],{"xmlns":58},[61,2189,2190,2202],{},[64,2191,2192,2194,2196,2198,2200],{},[79,2193,92],{"stretchy":91},[70,2195,119],{},[79,2197,81],{},[83,2199,85],{},[79,2201,112],{"stretchy":91},[121,2203,2204],{"encoding":123},"(M+1)",[42,2206,2208,2229],{"className":2207,"ariaHidden":129},[128],[42,2209,2211,2214,2217,2220,2223,2226],{"className":2210},[133],[42,2212],{"className":2213,"style":227},[137],[42,2215,92],{"className":2216},[231],[42,2218,119],{"className":2219,"style":147},[142,146],[42,2221],{"className":2222,"style":238},[212],[42,2224,81],{"className":2225},[191],[42,2227],{"className":2228,"style":238},[212],[42,2230,2232,2235,2238],{"className":2231},[133],[42,2233],{"className":2234,"style":227},[137],[42,2236,85],{"className":2237},[142],[42,2239,112],{"className":2240},[316],"-th number will necessarily be the same as one of the previous numbers, and since the recursive relationship is one-to-one, the subsequent sequence will repeat, leading to a period ",[42,2243,2245,2264],{"className":2244,"translate":46},[50],[42,2246,2248],{"className":2247},[54],[56,2249,2250],{"xmlns":58},[61,2251,2252,2261],{},[64,2253,2254,2256,2259],{},[70,2255,1477],{},[79,2257,2258],{},"≤",[70,2260,119],{},[121,2262,2263],{"encoding":123},"T \\leq M",[42,2265,2267,2286],{"className":2266,"ariaHidden":129},[128],[42,2268,2270,2274,2277,2280,2283],{"className":2269},[133],[42,2271],{"className":2272,"style":2273},[137],"height:0.8193em;vertical-align:-0.136em;",[42,2275,1477],{"className":2276,"style":1492},[142,146],[42,2278],{"className":2279,"style":213},[212],[42,2281,2258],{"className":2282},[217],[42,2284],{"className":2285,"style":213},[212],[42,2287,2289,2292],{"className":2288},[133],[42,2290],{"className":2291,"style":351},[137],[42,2293,119],{"className":2294,"style":147},[142,146],[15,2296,2297,2298,2550,2551,596,2579,2607,2608,596,2683,2756,2757,2810,2811,3050,3051,3079,3080,3108,3109,3457,3458,3565,3566,3594,3595,3646,3647,3675,3676,3704,3705,3765],{},"Next, it is important that the multiplier and modulus be coprime. This is easy to understand: consider ",[42,2299,2301,2362],{"className":2300,"translate":46},[50],[42,2302,2304],{"className":2303},[54],[56,2305,2306],{"xmlns":58},[61,2307,2308,2360],{},[64,2309,2310,2322,2324,2326,2328,2330,2336,2338,2340,2342,2344,2346,2354,2356,2358],{},[67,2311,2312,2314],{},[70,2313,72],{},[64,2315,2316,2318,2320],{},[70,2317,77],{},[79,2319,81],{},[83,2321,85],{},[79,2323,88],{},[79,2325,92],{"stretchy":91},[70,2327,95],{},[79,2329,98],{},[67,2331,2332,2334],{},[70,2333,72],{},[70,2335,77],{},[79,2337,81],{},[70,2339,109],{},[79,2341,112],{"stretchy":91},[212,2343],{},[212,2345],{"width":1245},[64,2347,2348,2350,2352],{},[70,2349,1250],{"mathvariant":115},[70,2351,1253],{"mathvariant":115},[70,2353,1256],{"mathvariant":115},[1258,2355,1260],{},[1258,2357,1260],{},[70,2359,119],{},[121,2361,1267],{"encoding":123},[42,2363,2365,2429,2450,2508,2526],{"className":2364,"ariaHidden":129},[128],[42,2366,2368,2371,2420,2423,2426],{"className":2367},[133],[42,2369],{"className":2370,"style":138},[137],[42,2372,2374,2377],{"className":2373},[142],[42,2375,72],{"className":2376,"style":147},[142,146],[42,2378,2380],{"className":2379},[151],[42,2381,2383,2412],{"className":2382},[155,156],[42,2384,2386,2409],{"className":2385},[160],[42,2387,2389],{"className":2388,"style":165},[164],[42,2390,2391,2394],{"style":168},[42,2392],{"className":2393,"style":173},[172],[42,2395,2397],{"className":2396},[177,178,179,180],[42,2398,2400,2403,2406],{"className":2399},[142,180],[42,2401,77],{"className":2402,"style":187},[142,146,180],[42,2404,81],{"className":2405},[191,180],[42,2407,85],{"className":2408},[142,180],[42,2410,199],{"className":2411},[198],[42,2413,2415],{"className":2414},[160],[42,2416,2418],{"className":2417,"style":206},[164],[42,2419],{},[42,2421],{"className":2422,"style":213},[212],[42,2424,88],{"className":2425},[217],[42,2427],{"className":2428,"style":213},[212],[42,2430,2432,2435,2438,2441,2444,2447],{"className":2431},[133],[42,2433],{"className":2434,"style":227},[137],[42,2436,92],{"className":2437},[231],[42,2439,95],{"className":2440},[142,146],[42,2442],{"className":2443,"style":238},[212],[42,2445,98],{"className":2446},[191],[42,2448],{"className":2449,"style":238},[212],[42,2451,2453,2456,2499,2502,2505],{"className":2452},[133],[42,2454],{"className":2455,"style":138},[137],[42,2457,2459,2462],{"className":2458},[142],[42,2460,72],{"className":2461,"style":147},[142,146],[42,2463,2465],{"className":2464},[151],[42,2466,2468,2491],{"className":2467},[155,156],[42,2469,2471,2488],{"className":2470},[160],[42,2472,2474],{"className":2473,"style":165},[164],[42,2475,2476,2479],{"style":168},[42,2477],{"className":2478,"style":173},[172],[42,2480,2482],{"className":2481},[177,178,179,180],[42,2483,2485],{"className":2484},[142,180],[42,2486,77],{"className":2487,"style":187},[142,146,180],[42,2489,199],{"className":2490},[198],[42,2492,2494],{"className":2493},[160],[42,2495,2497],{"className":2496,"style":206},[164],[42,2498],{},[42,2500],{"className":2501,"style":238},[212],[42,2503,81],{"className":2504},[191],[42,2506],{"className":2507,"style":238},[212],[42,2509,2511,2514,2517,2520,2523],{"className":2510},[133],[42,2512],{"className":2513,"style":227},[137],[42,2515,109],{"className":2516,"style":312},[142,146],[42,2518,112],{"className":2519},[316],[42,2521],{"className":2522},[212,1429],[42,2524],{"className":2525,"style":1433},[212],[42,2527,2529,2532,2541,2544,2547],{"className":2528},[133],[42,2530],{"className":2531,"style":1440},[137],[42,2533,2535],{"className":2534},[142],[42,2536,2538],{"className":2537},[142],[42,2539,1451],{"className":2540},[142,1450],[42,2542],{"className":2543,"style":1455},[212],[42,2545],{"className":2546,"style":1455},[212],[42,2548,119],{"className":2549,"style":147},[142,146],". If ",[42,2552,2554,2567],{"className":2553,"translate":46},[50],[42,2555,2557],{"className":2556},[54],[56,2558,2559],{"xmlns":58},[61,2560,2561,2565],{},[64,2562,2563],{},[70,2564,95],{},[121,2566,95],{"encoding":123},[42,2568,2570],{"className":2569,"ariaHidden":129},[128],[42,2571,2573,2576],{"className":2572},[133],[42,2574],{"className":2575,"style":351},[137],[42,2577,95],{"className":2578},[142,146],[42,2580,2582,2595],{"className":2581,"translate":46},[50],[42,2583,2585],{"className":2584},[54],[56,2586,2587],{"xmlns":58},[61,2588,2589,2593],{},[64,2590,2591],{},[70,2592,119],{},[121,2594,119],{"encoding":123},[42,2596,2598],{"className":2597,"ariaHidden":129},[128],[42,2599,2601,2604],{"className":2600},[133],[42,2602],{"className":2603,"style":351},[137],[42,2605,119],{"className":2606,"style":147},[142,146]," are not coprime, suppose ",[42,2609,2611,2634],{"className":2610,"translate":46},[50],[42,2612,2614],{"className":2613},[54],[56,2615,2616],{"xmlns":58},[61,2617,2618,2631],{},[64,2619,2620,2622,2624,2627,2629],{},[70,2621,95],{},[79,2623,88],{},[70,2625,2626],{},"a",[79,2628,98],{},[70,2630,1256],{},[121,2632,2633],{"encoding":123},"A = a * d",[42,2635,2637,2655,2674],{"className":2636,"ariaHidden":129},[128],[42,2638,2640,2643,2646,2649,2652],{"className":2639},[133],[42,2641],{"className":2642,"style":351},[137],[42,2644,95],{"className":2645},[142,146],[42,2647],{"className":2648,"style":213},[212],[42,2650,88],{"className":2651},[217],[42,2653],{"className":2654,"style":213},[212],[42,2656,2658,2662,2665,2668,2671],{"className":2657},[133],[42,2659],{"className":2660,"style":2661},[137],"height:0.4653em;",[42,2663,2626],{"className":2664},[142,146],[42,2666],{"className":2667,"style":238},[212],[42,2669,98],{"className":2670},[191],[42,2672],{"className":2673,"style":238},[212],[42,2675,2677,2680],{"className":2676},[133],[42,2678],{"className":2679,"style":1440},[137],[42,2681,1256],{"className":2682},[142,146],[42,2684,2686,2708],{"className":2685,"translate":46},[50],[42,2687,2689],{"className":2688},[54],[56,2690,2691],{"xmlns":58},[61,2692,2693,2705],{},[64,2694,2695,2697,2699,2701,2703],{},[70,2696,119],{},[79,2698,88],{},[70,2700,1250],{},[79,2702,98],{},[70,2704,1256],{},[121,2706,2707],{"encoding":123},"M = m * d",[42,2709,2711,2729,2747],{"className":2710,"ariaHidden":129},[128],[42,2712,2714,2717,2720,2723,2726],{"className":2713},[133],[42,2715],{"className":2716,"style":351},[137],[42,2718,119],{"className":2719,"style":147},[142,146],[42,2721],{"className":2722,"style":213},[212],[42,2724,88],{"className":2725},[217],[42,2727],{"className":2728,"style":213},[212],[42,2730,2732,2735,2738,2741,2744],{"className":2731},[133],[42,2733],{"className":2734,"style":2661},[137],[42,2736,1250],{"className":2737},[142,146],[42,2739],{"className":2740,"style":238},[212],[42,2742,98],{"className":2743},[191],[42,2745],{"className":2746,"style":238},[212],[42,2748,2750,2753],{"className":2749},[133],[42,2751],{"className":2752,"style":1440},[137],[42,2754,1256],{"className":2755},[142,146]," where ",[42,2758,2760,2779],{"className":2759,"translate":46},[50],[42,2761,2763],{"className":2762},[54],[56,2764,2765],{"xmlns":58},[61,2766,2767,2776],{},[64,2768,2769,2771,2774],{},[70,2770,1256],{},[79,2772,2773],{},">",[83,2775,85],{},[121,2777,2778],{"encoding":123},"d > 1",[42,2780,2782,2801],{"className":2781,"ariaHidden":129},[128],[42,2783,2785,2789,2792,2795,2798],{"className":2784},[133],[42,2786],{"className":2787,"style":2788},[137],"height:0.7335em;vertical-align:-0.0391em;",[42,2790,1256],{"className":2791},[142,146],[42,2793],{"className":2794,"style":213},[212],[42,2796,2773],{"className":2797},[217],[42,2799],{"className":2800,"style":213},[212],[42,2802,2804,2807],{"className":2803},[133],[42,2805],{"className":2806,"style":708},[137],[42,2808,85],{"className":2809},[142],". Expanding the recursive formula yields ",[42,2812,2814,2862],{"className":2813,"translate":46},[50],[42,2815,2817],{"className":2816},[54],[56,2818,2819],{"xmlns":58},[61,2820,2821,2859],{},[64,2822,2823,2835,2837,2839,2841,2847,2849,2851,2853,2855,2857],{},[67,2824,2825,2827],{},[70,2826,72],{},[64,2828,2829,2831,2833],{},[70,2830,77],{},[79,2832,81],{},[83,2834,85],{},[79,2836,88],{},[70,2838,95],{},[79,2840,98],{},[67,2842,2843,2845],{},[70,2844,72],{},[70,2846,77],{},[79,2848,81],{},[70,2850,109],{},[79,2852,81],{},[70,2854,1513],{},[79,2856,98],{},[70,2858,119],{},[121,2860,2861],{"encoding":123},"N_{j+1} = A * N_{j} + B + k * M",[42,2863,2865,2929,2947,3005,3023,3041],{"className":2864,"ariaHidden":129},[128],[42,2866,2868,2871,2920,2923,2926],{"className":2867},[133],[42,2869],{"className":2870,"style":138},[137],[42,2872,2874,2877],{"className":2873},[142],[42,2875,72],{"className":2876,"style":147},[142,146],[42,2878,2880],{"className":2879},[151],[42,2881,2883,2912],{"className":2882},[155,156],[42,2884,2886,2909],{"className":2885},[160],[42,2887,2889],{"className":2888,"style":165},[164],[42,2890,2891,2894],{"style":168},[42,2892],{"className":2893,"style":173},[172],[42,2895,2897],{"className":2896},[177,178,179,180],[42,2898,2900,2903,2906],{"className":2899},[142,180],[42,2901,77],{"className":2902,"style":187},[142,146,180],[42,2904,81],{"className":2905},[191,180],[42,2907,85],{"className":2908},[142,180],[42,2910,199],{"className":2911},[198],[42,2913,2915],{"className":2914},[160],[42,2916,2918],{"className":2917,"style":206},[164],[42,2919],{},[42,2921],{"className":2922,"style":213},[212],[42,2924,88],{"className":2925},[217],[42,2927],{"className":2928,"style":213},[212],[42,2930,2932,2935,2938,2941,2944],{"className":2931},[133],[42,2933],{"className":2934,"style":351},[137],[42,2936,95],{"className":2937},[142,146],[42,2939],{"className":2940,"style":238},[212],[42,2942,98],{"className":2943},[191],[42,2945],{"className":2946,"style":238},[212],[42,2948,2950,2953,2996,2999,3002],{"className":2949},[133],[42,2951],{"className":2952,"style":138},[137],[42,2954,2956,2959],{"className":2955},[142],[42,2957,72],{"className":2958,"style":147},[142,146],[42,2960,2962],{"className":2961},[151],[42,2963,2965,2988],{"className":2964},[155,156],[42,2966,2968,2985],{"className":2967},[160],[42,2969,2971],{"className":2970,"style":165},[164],[42,2972,2973,2976],{"style":168},[42,2974],{"className":2975,"style":173},[172],[42,2977,2979],{"className":2978},[177,178,179,180],[42,2980,2982],{"className":2981},[142,180],[42,2983,77],{"className":2984,"style":187},[142,146,180],[42,2986,199],{"className":2987},[198],[42,2989,2991],{"className":2990},[160],[42,2992,2994],{"className":2993,"style":206},[164],[42,2995],{},[42,2997],{"className":2998,"style":238},[212],[42,3000,81],{"className":3001},[191],[42,3003],{"className":3004,"style":238},[212],[42,3006,3008,3011,3014,3017,3020],{"className":3007},[133],[42,3009],{"className":3010,"style":689},[137],[42,3012,109],{"className":3013,"style":312},[142,146],[42,3015],{"className":3016,"style":238},[212],[42,3018,81],{"className":3019},[191],[42,3021],{"className":3022,"style":238},[212],[42,3024,3026,3029,3032,3035,3038],{"className":3025},[133],[42,3027],{"className":3028,"style":1440},[137],[42,3030,1513],{"className":3031,"style":1572},[142,146],[42,3033],{"className":3034,"style":238},[212],[42,3036,98],{"className":3037},[191],[42,3039],{"className":3040,"style":238},[212],[42,3042,3044,3047],{"className":3043},[133],[42,3045],{"className":3046,"style":351},[137],[42,3048,119],{"className":3049,"style":147},[142,146],". Since ",[42,3052,3054,3067],{"className":3053,"translate":46},[50],[42,3055,3057],{"className":3056},[54],[56,3058,3059],{"xmlns":58},[61,3060,3061,3065],{},[64,3062,3063],{},[70,3064,109],{},[121,3066,109],{"encoding":123},[42,3068,3070],{"className":3069,"ariaHidden":129},[128],[42,3071,3073,3076],{"className":3072},[133],[42,3074],{"className":3075,"style":351},[137],[42,3077,109],{"className":3078,"style":312},[142,146]," is constant, it can be treated as a shift, and by subtracting ",[42,3081,3083,3096],{"className":3082,"translate":46},[50],[42,3084,3086],{"className":3085},[54],[56,3087,3088],{"xmlns":58},[61,3089,3090,3094],{},[64,3091,3092],{},[70,3093,109],{},[121,3095,109],{"encoding":123},[42,3097,3099],{"className":3098,"ariaHidden":129},[128],[42,3100,3102,3105],{"className":3101},[133],[42,3103],{"className":3104,"style":351},[137],[42,3106,109],{"className":3107,"style":312},[142,146]," from both sides, we get ",[42,3110,3112,3184],{"className":3111,"translate":46},[50],[42,3113,3115],{"className":3114},[54],[56,3116,3117],{"xmlns":58},[61,3118,3119,3181],{},[64,3120,3121,3133,3135,3137,3139,3141,3143,3145,3147,3153,3155,3157,3159,3161,3163,3165,3167,3169,3171,3173,3175,3177,3179],{},[67,3122,3123,3125],{},[70,3124,72],{},[64,3126,3127,3129,3131],{},[70,3128,77],{},[79,3130,81],{},[83,3132,85],{},[79,3134,674],{},[70,3136,109],{},[79,3138,88],{},[79,3140,92],{"stretchy":91},[70,3142,2626],{},[79,3144,98],{},[79,3146,92],{"stretchy":91},[67,3148,3149,3151],{},[70,3150,72],{},[70,3152,77],{},[79,3154,674],{},[70,3156,109],{},[79,3158,112],{"stretchy":91},[79,3160,81],{},[70,3162,2626],{},[79,3164,98],{},[70,3166,109],{},[79,3168,81],{},[70,3170,1513],{},[79,3172,98],{},[70,3174,1250],{},[79,3176,112],{"stretchy":91},[79,3178,98],{},[70,3180,1256],{},[121,3182,3183],{"encoding":123},"N_{j+1} - B = (a * (N_{j} - B) + a * B + k * m) * d",[42,3185,3187,3251,3269,3290,3352,3373,3391,3409,3427,3448],{"className":3186,"ariaHidden":129},[128],[42,3188,3190,3193,3242,3245,3248],{"className":3189},[133],[42,3191],{"className":3192,"style":138},[137],[42,3194,3196,3199],{"className":3195},[142],[42,3197,72],{"className":3198,"style":147},[142,146],[42,3200,3202],{"className":3201},[151],[42,3203,3205,3234],{"className":3204},[155,156],[42,3206,3208,3231],{"className":3207},[160],[42,3209,3211],{"className":3210,"style":165},[164],[42,3212,3213,3216],{"style":168},[42,3214],{"className":3215,"style":173},[172],[42,3217,3219],{"className":3218},[177,178,179,180],[42,3220,3222,3225,3228],{"className":3221},[142,180],[42,3223,77],{"className":3224,"style":187},[142,146,180],[42,3226,81],{"className":3227},[191,180],[42,3229,85],{"className":3230},[142,180],[42,3232,199],{"className":3233},[198],[42,3235,3237],{"className":3236},[160],[42,3238,3240],{"className":3239,"style":206},[164],[42,3241],{},[42,3243],{"className":3244,"style":238},[212],[42,3246,674],{"className":3247},[191],[42,3249],{"className":3250,"style":238},[212],[42,3252,3254,3257,3260,3263,3266],{"className":3253},[133],[42,3255],{"className":3256,"style":351},[137],[42,3258,109],{"className":3259,"style":312},[142,146],[42,3261],{"className":3262,"style":213},[212],[42,3264,88],{"className":3265},[217],[42,3267],{"className":3268,"style":213},[212],[42,3270,3272,3275,3278,3281,3284,3287],{"className":3271},[133],[42,3273],{"className":3274,"style":227},[137],[42,3276,92],{"className":3277},[231],[42,3279,2626],{"className":3280},[142,146],[42,3282],{"className":3283,"style":238},[212],[42,3285,98],{"className":3286},[191],[42,3288],{"className":3289,"style":238},[212],[42,3291,3293,3297,3300,3343,3346,3349],{"className":3292},[133],[42,3294],{"className":3295,"style":3296},[137],"height:1.0361em;vertical-align:-0.2861em;",[42,3298,92],{"className":3299},[231],[42,3301,3303,3306],{"className":3302},[142],[42,3304,72],{"className":3305,"style":147},[142,146],[42,3307,3309],{"className":3308},[151],[42,3310,3312,3335],{"className":3311},[155,156],[42,3313,3315,3332],{"className":3314},[160],[42,3316,3318],{"className":3317,"style":165},[164],[42,3319,3320,3323],{"style":168},[42,3321],{"className":3322,"style":173},[172],[42,3324,3326],{"className":3325},[177,178,179,180],[42,3327,3329],{"className":3328},[142,180],[42,3330,77],{"className":3331,"style":187},[142,146,180],[42,3333,199],{"className":3334},[198],[42,3336,3338],{"className":3337},[160],[42,3339,3341],{"className":3340,"style":206},[164],[42,3342],{},[42,3344],{"className":3345,"style":238},[212],[42,3347,674],{"className":3348},[191],[42,3350],{"className":3351,"style":238},[212],[42,3353,3355,3358,3361,3364,3367,3370],{"className":3354},[133],[42,3356],{"className":3357,"style":227},[137],[42,3359,109],{"className":3360,"style":312},[142,146],[42,3362,112],{"className":3363},[316],[42,3365],{"className":3366,"style":238},[212],[42,3368,81],{"className":3369},[191],[42,3371],{"className":3372,"style":238},[212],[42,3374,3376,3379,3382,3385,3388],{"className":3375},[133],[42,3377],{"className":3378,"style":2661},[137],[42,3380,2626],{"className":3381},[142,146],[42,3383],{"className":3384,"style":238},[212],[42,3386,98],{"className":3387},[191],[42,3389],{"className":3390,"style":238},[212],[42,3392,3394,3397,3400,3403,3406],{"className":3393},[133],[42,3395],{"className":3396,"style":689},[137],[42,3398,109],{"className":3399,"style":312},[142,146],[42,3401],{"className":3402,"style":238},[212],[42,3404,81],{"className":3405},[191],[42,3407],{"className":3408,"style":238},[212],[42,3410,3412,3415,3418,3421,3424],{"className":3411},[133],[42,3413],{"className":3414,"style":1440},[137],[42,3416,1513],{"className":3417,"style":1572},[142,146],[42,3419],{"className":3420,"style":238},[212],[42,3422,98],{"className":3423},[191],[42,3425],{"className":3426,"style":238},[212],[42,3428,3430,3433,3436,3439,3442,3445],{"className":3429},[133],[42,3431],{"className":3432,"style":227},[137],[42,3434,1250],{"className":3435},[142,146],[42,3437,112],{"className":3438},[316],[42,3440],{"className":3441,"style":238},[212],[42,3443,98],{"className":3444},[191],[42,3446],{"className":3447,"style":238},[212],[42,3449,3451,3454],{"className":3450},[133],[42,3452],{"className":3453,"style":1440},[137],[42,3455,1256],{"className":3456},[142,146],". This shows that all values of ",[42,3459,3461,3489],{"className":3460,"translate":46},[50],[42,3462,3464],{"className":3463},[54],[56,3465,3466],{"xmlns":58},[61,3467,3468,3486],{},[64,3469,3470,3482,3484],{},[67,3471,3472,3474],{},[70,3473,72],{},[64,3475,3476,3478,3480],{},[70,3477,77],{},[79,3479,81],{},[83,3481,85],{},[79,3483,674],{},[70,3485,109],{},[121,3487,3488],{"encoding":123},"N_{j+1} - B",[42,3490,3492,3556],{"className":3491,"ariaHidden":129},[128],[42,3493,3495,3498,3547,3550,3553],{"className":3494},[133],[42,3496],{"className":3497,"style":138},[137],[42,3499,3501,3504],{"className":3500},[142],[42,3502,72],{"className":3503,"style":147},[142,146],[42,3505,3507],{"className":3506},[151],[42,3508,3510,3539],{"className":3509},[155,156],[42,3511,3513,3536],{"className":3512},[160],[42,3514,3516],{"className":3515,"style":165},[164],[42,3517,3518,3521],{"style":168},[42,3519],{"className":3520,"style":173},[172],[42,3522,3524],{"className":3523},[177,178,179,180],[42,3525,3527,3530,3533],{"className":3526},[142,180],[42,3528,77],{"className":3529,"style":187},[142,146,180],[42,3531,81],{"className":3532},[191,180],[42,3534,85],{"className":3535},[142,180],[42,3537,199],{"className":3538},[198],[42,3540,3542],{"className":3541},[160],[42,3543,3545],{"className":3544,"style":206},[164],[42,3546],{},[42,3548],{"className":3549,"style":238},[212],[42,3551,674],{"className":3552},[191],[42,3554],{"className":3555,"style":238},[212],[42,3557,3559,3562],{"className":3558},[133],[42,3560],{"className":3561,"style":351},[137],[42,3563,109],{"className":3564,"style":312},[142,146]," contain the factor ",[42,3567,3569,3582],{"className":3568,"translate":46},[50],[42,3570,3572],{"className":3571},[54],[56,3573,3574],{"xmlns":58},[61,3575,3576,3580],{},[64,3577,3578],{},[70,3579,1256],{},[121,3581,1256],{"encoding":123},[42,3583,3585],{"className":3584,"ariaHidden":129},[128],[42,3586,3588,3591],{"className":3587},[133],[42,3589],{"className":3590,"style":1440},[137],[42,3592,1256],{"className":3593},[142,146],". For example, if ",[42,3596,3598,3616],{"className":3597,"translate":46},[50],[42,3599,3601],{"className":3600},[54],[56,3602,3603],{"xmlns":58},[61,3604,3605,3613],{},[64,3606,3607,3609,3611],{},[70,3608,1256],{},[79,3610,88],{},[83,3612,1074],{},[121,3614,3615],{"encoding":123},"d = 2",[42,3617,3619,3637],{"className":3618,"ariaHidden":129},[128],[42,3620,3622,3625,3628,3631,3634],{"className":3621},[133],[42,3623],{"className":3624,"style":1440},[137],[42,3626,1256],{"className":3627},[142,146],[42,3629],{"className":3630,"style":213},[212],[42,3632,88],{"className":3633},[217],[42,3635],{"className":3636,"style":213},[212],[42,3638,3640,3643],{"className":3639},[133],[42,3641],{"className":3642,"style":708},[137],[42,3644,1074],{"className":3645},[142],", the sequence of ",[42,3648,3650,3663],{"className":3649,"translate":46},[50],[42,3651,3653],{"className":3652},[54],[56,3654,3655],{"xmlns":58},[61,3656,3657,3661],{},[64,3658,3659],{},[70,3660,119],{},[121,3662,119],{"encoding":123},[42,3664,3666],{"className":3665,"ariaHidden":129},[128],[42,3667,3669,3672],{"className":3668},[133],[42,3670],{"className":3671,"style":351},[137],[42,3673,119],{"className":3674,"style":147},[142,146]," possible numbers will only yield even numbers plus the shift ",[42,3677,3679,3692],{"className":3678,"translate":46},[50],[42,3680,3682],{"className":3681},[54],[56,3683,3684],{"xmlns":58},[61,3685,3686,3690],{},[64,3687,3688],{},[70,3689,109],{},[121,3691,109],{"encoding":123},[42,3693,3695],{"className":3694,"ariaHidden":129},[128],[42,3696,3698,3701],{"className":3697},[133],[42,3699],{"className":3700,"style":351},[137],[42,3702,109],{"className":3703,"style":312},[142,146],", meaning the period ",[42,3706,3708,3731],{"className":3707,"translate":46},[50],[42,3709,3711],{"className":3710},[54],[56,3712,3713],{"xmlns":58},[61,3714,3715,3728],{},[64,3716,3717,3719,3721,3723,3726],{},[70,3718,1477],{},[79,3720,2258],{},[70,3722,119],{},[70,3724,3725],{"mathvariant":115},"\u002F",[83,3727,1074],{},[121,3729,3730],{"encoding":123},"T \\leq M \u002F 2",[42,3732,3734,3752],{"className":3733,"ariaHidden":129},[128],[42,3735,3737,3740,3743,3746,3749],{"className":3736},[133],[42,3738],{"className":3739,"style":2273},[137],[42,3741,1477],{"className":3742,"style":1492},[142,146],[42,3744],{"className":3745,"style":213},[212],[42,3747,2258],{"className":3748},[217],[42,3750],{"className":3751,"style":213},[212],[42,3753,3755,3758,3761],{"className":3754},[133],[42,3756],{"className":3757,"style":227},[137],[42,3759,119],{"className":3760,"style":147},[142,146],[42,3762,3764],{"className":3763},[142],"\u002F2",". This disrupts the uniform distribution of the sequence.",[15,3767,3768,3769,445,3821,474,3873,3965,3966,4094,4095,445,4147,474,4199,3965,4290,4373],{},"A few simple examples can illustrate this. Suppose ",[42,3770,3772,3791],{"className":3771,"translate":46},[50],[42,3773,3775],{"className":3774},[54],[56,3776,3777],{"xmlns":58},[61,3778,3779,3788],{},[64,3780,3781,3783,3785],{},[70,3782,95],{},[79,3784,88],{},[83,3786,3787],{},"3",[121,3789,3790],{"encoding":123},"A = 3",[42,3792,3794,3812],{"className":3793,"ariaHidden":129},[128],[42,3795,3797,3800,3803,3806,3809],{"className":3796},[133],[42,3798],{"className":3799,"style":351},[137],[42,3801,95],{"className":3802},[142,146],[42,3804],{"className":3805,"style":213},[212],[42,3807,88],{"className":3808},[217],[42,3810],{"className":3811,"style":213},[212],[42,3813,3815,3818],{"className":3814},[133],[42,3816],{"className":3817,"style":708},[137],[42,3819,3787],{"className":3820},[142],[42,3822,3824,3843],{"className":3823,"translate":46},[50],[42,3825,3827],{"className":3826},[54],[56,3828,3829],{"xmlns":58},[61,3830,3831,3840],{},[64,3832,3833,3835,3837],{},[70,3834,119],{},[79,3836,88],{},[83,3838,3839],{},"5",[121,3841,3842],{"encoding":123},"M = 5",[42,3844,3846,3864],{"className":3845,"ariaHidden":129},[128],[42,3847,3849,3852,3855,3858,3861],{"className":3848},[133],[42,3850],{"className":3851,"style":351},[137],[42,3853,119],{"className":3854,"style":147},[142,146],[42,3856],{"className":3857,"style":213},[212],[42,3859,88],{"className":3860},[217],[42,3862],{"className":3863,"style":213},[212],[42,3865,3867,3870],{"className":3866},[133],[42,3868],{"className":3869,"style":708},[137],[42,3871,3839],{"className":3872},[142],[42,3874,3876,3898],{"className":3875,"translate":46},[50],[42,3877,3879],{"className":3878},[54],[56,3880,3881],{"xmlns":58},[61,3882,3883,3895],{},[64,3884,3885,3891,3893],{},[67,3886,3887,3889],{},[70,3888,72],{},[83,3890,739],{},[79,3892,88],{},[83,3894,1074],{},[121,3896,3897],{"encoding":123},"N_0 = 2",[42,3899,3901,3956],{"className":3900,"ariaHidden":129},[128],[42,3902,3904,3907,3947,3950,3953],{"className":3903},[133],[42,3905],{"className":3906,"style":918},[137],[42,3908,3910,3913],{"className":3909},[142],[42,3911,72],{"className":3912,"style":147},[142,146],[42,3914,3916],{"className":3915},[151],[42,3917,3919,3939],{"className":3918},[155,156],[42,3920,3922,3936],{"className":3921},[160],[42,3923,3925],{"className":3924,"style":937},[164],[42,3926,3927,3930],{"style":168},[42,3928],{"className":3929,"style":173},[172],[42,3931,3933],{"className":3932},[177,178,179,180],[42,3934,739],{"className":3935},[142,180],[42,3937,199],{"className":3938},[198],[42,3940,3942],{"className":3941},[160],[42,3943,3945],{"className":3944,"style":961},[164],[42,3946],{},[42,3948],{"className":3949,"style":213},[212],[42,3951,88],{"className":3952},[217],[42,3954],{"className":3955,"style":213},[212],[42,3957,3959,3962],{"className":3958},[133],[42,3960],{"className":3961,"style":708},[137],[42,3963,1074],{"className":3964},[142],", the sequence would be ",[42,3967,3969,4017],{"className":3968,"translate":46},[50],[42,3970,3972],{"className":3971},[54],[56,3973,3974],{"xmlns":58},[61,3975,3976,4014],{},[64,3977,3978,3981,3983,3986,3988,3990,3992,3994,3996,3998,4000,4002,4004,4006,4009,4011],{},[79,3979,3980],{"stretchy":91},"{",[83,3982,1074],{},[79,3984,3985],{"separator":129},",",[83,3987,85],{},[79,3989,3985],{"separator":129},[83,3991,3787],{},[79,3993,3985],{"separator":129},[83,3995,734],{},[79,3997,3985],{"separator":129},[83,3999,1074],{},[79,4001,3985],{"separator":129},[83,4003,85],{},[79,4005,3985],{"separator":129},[79,4007,4008],{},"…",[1258,4010,1260],{},[79,4012,4013],{"stretchy":91},"}",[121,4015,4016],{"encoding":123},"\\{2, 1, 3, 4, 2, 1, \\dots\\}",[42,4018,4020],{"className":4019,"ariaHidden":129},[128],[42,4021,4023,4026,4029,4032,4036,4039,4042,4045,4048,4051,4054,4057,4060,4063,4066,4069,4072,4075,4078,4081,4084,4088,4091],{"className":4022},[133],[42,4024],{"className":4025,"style":227},[137],[42,4027,3980],{"className":4028},[231],[42,4030,1074],{"className":4031},[142],[42,4033,3985],{"className":4034},[4035],"mpunct",[42,4037],{"className":4038,"style":1455},[212],[42,4040,85],{"className":4041},[142],[42,4043,3985],{"className":4044},[4035],[42,4046],{"className":4047,"style":1455},[212],[42,4049,3787],{"className":4050},[142],[42,4052,3985],{"className":4053},[4035],[42,4055],{"className":4056,"style":1455},[212],[42,4058,734],{"className":4059},[142],[42,4061,3985],{"className":4062},[4035],[42,4064],{"className":4065,"style":1455},[212],[42,4067,1074],{"className":4068},[142],[42,4070,3985],{"className":4071},[4035],[42,4073],{"className":4074,"style":1455},[212],[42,4076,85],{"className":4077},[142],[42,4079,3985],{"className":4080},[4035],[42,4082],{"className":4083,"style":1455},[212],[42,4085,4008],{"className":4086},[4087],"minner",[42,4089],{"className":4090,"style":1455},[212],[42,4092,4013],{"className":4093},[316],", with a period of 5. If we set ",[42,4096,4098,4117],{"className":4097,"translate":46},[50],[42,4099,4101],{"className":4100},[54],[56,4102,4103],{"xmlns":58},[61,4104,4105,4114],{},[64,4106,4107,4109,4111],{},[70,4108,95],{},[79,4110,88],{},[83,4112,4113],{},"6",[121,4115,4116],{"encoding":123},"A = 6",[42,4118,4120,4138],{"className":4119,"ariaHidden":129},[128],[42,4121,4123,4126,4129,4132,4135],{"className":4122},[133],[42,4124],{"className":4125,"style":351},[137],[42,4127,95],{"className":4128},[142,146],[42,4130],{"className":4131,"style":213},[212],[42,4133,88],{"className":4134},[217],[42,4136],{"className":4137,"style":213},[212],[42,4139,4141,4144],{"className":4140},[133],[42,4142],{"className":4143,"style":708},[137],[42,4145,4113],{"className":4146},[142],[42,4148,4150,4169],{"className":4149,"translate":46},[50],[42,4151,4153],{"className":4152},[54],[56,4154,4155],{"xmlns":58},[61,4156,4157,4166],{},[64,4158,4159,4161,4163],{},[70,4160,119],{},[79,4162,88],{},[83,4164,4165],{},"10",[121,4167,4168],{"encoding":123},"M = 10",[42,4170,4172,4190],{"className":4171,"ariaHidden":129},[128],[42,4173,4175,4178,4181,4184,4187],{"className":4174},[133],[42,4176],{"className":4177,"style":351},[137],[42,4179,119],{"className":4180,"style":147},[142,146],[42,4182],{"className":4183,"style":213},[212],[42,4185,88],{"className":4186},[217],[42,4188],{"className":4189,"style":213},[212],[42,4191,4193,4196],{"className":4192},[133],[42,4194],{"className":4195,"style":708},[137],[42,4197,4165],{"className":4198},[142],[42,4200,4202,4223],{"className":4201,"translate":46},[50],[42,4203,4205],{"className":4204},[54],[56,4206,4207],{"xmlns":58},[61,4208,4209,4221],{},[64,4210,4211,4217,4219],{},[67,4212,4213,4215],{},[70,4214,72],{},[83,4216,739],{},[79,4218,88],{},[83,4220,1074],{},[121,4222,3897],{"encoding":123},[42,4224,4226,4281],{"className":4225,"ariaHidden":129},[128],[42,4227,4229,4232,4272,4275,4278],{"className":4228},[133],[42,4230],{"className":4231,"style":918},[137],[42,4233,4235,4238],{"className":4234},[142],[42,4236,72],{"className":4237,"style":147},[142,146],[42,4239,4241],{"className":4240},[151],[42,4242,4244,4264],{"className":4243},[155,156],[42,4245,4247,4261],{"className":4246},[160],[42,4248,4250],{"className":4249,"style":937},[164],[42,4251,4252,4255],{"style":168},[42,4253],{"className":4254,"style":173},[172],[42,4256,4258],{"className":4257},[177,178,179,180],[42,4259,739],{"className":4260},[142,180],[42,4262,199],{"className":4263},[198],[42,4265,4267],{"className":4266},[160],[42,4268,4270],{"className":4269,"style":961},[164],[42,4271],{},[42,4273],{"className":4274,"style":213},[212],[42,4276,88],{"className":4277},[217],[42,4279],{"className":4280,"style":213},[212],[42,4282,4284,4287],{"className":4283},[133],[42,4285],{"className":4286,"style":708},[137],[42,4288,1074],{"className":4289},[142],[42,4291,4293,4325],{"className":4292,"translate":46},[50],[42,4294,4296],{"className":4295},[54],[56,4297,4298],{"xmlns":58},[61,4299,4300,4322],{},[64,4301,4302,4304,4306,4308,4310,4312,4314,4316,4318,4320],{},[79,4303,3980],{"stretchy":91},[83,4305,1074],{},[79,4307,3985],{"separator":129},[83,4309,1074],{},[79,4311,3985],{"separator":129},[83,4313,1074],{},[79,4315,3985],{"separator":129},[79,4317,4008],{},[1258,4319,1260],{},[79,4321,4013],{"stretchy":91},[121,4323,4324],{"encoding":123},"\\{2, 2, 2, \\dots\\}",[42,4326,4328],{"className":4327,"ariaHidden":129},[128],[42,4329,4331,4334,4337,4340,4343,4346,4349,4352,4355,4358,4361,4364,4367,4370],{"className":4330},[133],[42,4332],{"className":4333,"style":227},[137],[42,4335,3980],{"className":4336},[231],[42,4338,1074],{"className":4339},[142],[42,4341,3985],{"className":4342},[4035],[42,4344],{"className":4345,"style":1455},[212],[42,4347,1074],{"className":4348},[142],[42,4350,3985],{"className":4351},[4035],[42,4353],{"className":4354,"style":1455},[212],[42,4356,1074],{"className":4357},[142],[42,4359,3985],{"className":4360},[4035],[42,4362],{"className":4363,"style":1455},[212],[42,4365,4008],{"className":4366},[4087],[42,4368],{"className":4369,"style":1455},[212],[42,4371,4013],{"className":4372},[316],", with a period of 1.",[15,4375,4376,4377,4405],{},"Returning to the initial question, how does the LCG achieve uniform distribution? As seen from the discussion above, with carefully chosen parameters, it is possible to construct a sequence with a period of ",[42,4378,4380,4393],{"className":4379,"translate":46},[50],[42,4381,4383],{"className":4382},[54],[56,4384,4385],{"xmlns":58},[61,4386,4387,4391],{},[64,4388,4389],{},[70,4390,119],{},[121,4392,119],{"encoding":123},[42,4394,4396],{"className":4395,"ariaHidden":129},[128],[42,4397,4399,4402],{"className":4398},[133],[42,4400],{"className":4401,"style":351},[137],[42,4403,119],{"className":4404,"style":147},[142,146],", where each number appears exactly once in each cycle. This ensures that the frequency of appearance for each number in the sequence is the same, thus satisfying the uniform distribution requirement.",[10,4407,4409],{"id":4408},"solving-linear-congruences","Solving Linear Congruences",[15,4411,4412],{},"Having understood the relationship between random distribution and the linear congruential method, let's extend our knowledge to learn how to solve linear congruences. This is not just a theoretical mathematical problem; it has numerous practical applications. For instance:",[19,4414,4415],{},[15,4416,4417,4418,596,4446,4474,4475,596,4505,4536,4537,4565,4566,4594,4595,4565,4623,4652,4653,4682],{},"On a circular track, two frogs, ",[42,4419,4421,4434],{"className":4420,"translate":46},[50],[42,4422,4424],{"className":4423},[54],[56,4425,4426],{"xmlns":58},[61,4427,4428,4432],{},[64,4429,4430],{},[70,4431,95],{},[121,4433,95],{"encoding":123},[42,4435,4437],{"className":4436,"ariaHidden":129},[128],[42,4438,4440,4443],{"className":4439},[133],[42,4441],{"className":4442,"style":351},[137],[42,4444,95],{"className":4445},[142,146],[42,4447,4449,4462],{"className":4448,"translate":46},[50],[42,4450,4452],{"className":4451},[54],[56,4453,4454],{"xmlns":58},[61,4455,4456,4460],{},[64,4457,4458],{},[70,4459,109],{},[121,4461,109],{"encoding":123},[42,4463,4465],{"className":4464,"ariaHidden":129},[128],[42,4466,4468,4471],{"className":4467},[133],[42,4469],{"className":4470,"style":351},[137],[42,4472,109],{"className":4473,"style":312},[142,146],", start from positions ",[42,4476,4478,4492],{"className":4477,"translate":46},[50],[42,4479,4481],{"className":4480},[54],[56,4482,4483],{"xmlns":58},[61,4484,4485,4490],{},[64,4486,4487],{},[70,4488,4489],{},"x",[121,4491,4489],{"encoding":123},[42,4493,4495],{"className":4494,"ariaHidden":129},[128],[42,4496,4498,4502],{"className":4497},[133],[42,4499],{"className":4500,"style":4501},[137],"height:0.4306em;",[42,4503,4489],{"className":4504},[142,146],[42,4506,4508,4522],{"className":4507,"translate":46},[50],[42,4509,4511],{"className":4510},[54],[56,4512,4513],{"xmlns":58},[61,4514,4515,4520],{},[64,4516,4517],{},[70,4518,4519],{},"y",[121,4521,4519],{"encoding":123},[42,4523,4525],{"className":4524,"ariaHidden":129},[128],[42,4526,4528,4532],{"className":4527},[133],[42,4529],{"className":4530,"style":4531},[137],"height:0.625em;vertical-align:-0.1944em;",[42,4533,4519],{"className":4534,"style":4535},[142,146],"margin-right:0.03588em;"," respectively, with ",[42,4538,4540,4553],{"className":4539,"translate":46},[50],[42,4541,4543],{"className":4542},[54],[56,4544,4545],{"xmlns":58},[61,4546,4547,4551],{},[64,4548,4549],{},[70,4550,95],{},[121,4552,95],{"encoding":123},[42,4554,4556],{"className":4555,"ariaHidden":129},[128],[42,4557,4559,4562],{"className":4558},[133],[42,4560],{"className":4561,"style":351},[137],[42,4563,95],{"className":4564},[142,146]," jumping ",[42,4567,4569,4582],{"className":4568,"translate":46},[50],[42,4570,4572],{"className":4571},[54],[56,4573,4574],{"xmlns":58},[61,4575,4576,4580],{},[64,4577,4578],{},[70,4579,1250],{},[121,4581,1250],{"encoding":123},[42,4583,4585],{"className":4584,"ariaHidden":129},[128],[42,4586,4588,4591],{"className":4587},[133],[42,4589],{"className":4590,"style":4501},[137],[42,4592,1250],{"className":4593},[142,146]," units and ",[42,4596,4598,4611],{"className":4597,"translate":46},[50],[42,4599,4601],{"className":4600},[54],[56,4602,4603],{"xmlns":58},[61,4604,4605,4609],{},[64,4606,4607],{},[70,4608,109],{},[121,4610,109],{"encoding":123},[42,4612,4614],{"className":4613,"ariaHidden":129},[128],[42,4615,4617,4620],{"className":4616},[133],[42,4618],{"className":4619,"style":351},[137],[42,4621,109],{"className":4622,"style":312},[142,146],[42,4624,4626,4640],{"className":4625,"translate":46},[50],[42,4627,4629],{"className":4628},[54],[56,4630,4631],{"xmlns":58},[61,4632,4633,4638],{},[64,4634,4635],{},[70,4636,4637],{},"n",[121,4639,4637],{"encoding":123},[42,4641,4643],{"className":4642,"ariaHidden":129},[128],[42,4644,4646,4649],{"className":4645},[133],[42,4647],{"className":4648,"style":4501},[137],[42,4650,4637],{"className":4651},[142,146]," units each time. The total circumference of the track is ",[42,4654,4656,4670],{"className":4655,"translate":46},[50],[42,4657,4659],{"className":4658},[54],[56,4660,4661],{"xmlns":58},[61,4662,4663,4668],{},[64,4664,4665],{},[70,4666,4667],{},"L",[121,4669,4667],{"encoding":123},[42,4671,4673],{"className":4672,"ariaHidden":129},[128],[42,4674,4676,4679],{"className":4675},[133],[42,4677],{"className":4678,"style":351},[137],[42,4680,4667],{"className":4681},[142,146],". If both frogs start at the same time, they are said to meet if they land on the same point. The question is, after how many jumps will the two frogs meet at the earliest?",[15,4684,4685],{},"Based on these conditions, we can formulate the following equation:",[42,4687,4689],{"className":4688,"translate":46},[45],[42,4690,4692,4757],{"className":4691,"translate":46},[50],[42,4693,4695],{"className":4694},[54],[56,4696,4697],{"xmlns":58,"display":59},[61,4698,4699,4754],{},[64,4700,4701,4703,4705,4707,4709,4711,4713,4715,4718,4720,4722,4724,4726,4728,4730,4732,4734,4737,4739,4747,4750,4752],{},[79,4702,92],{"stretchy":91},[70,4704,4489],{},[79,4706,81],{},[70,4708,1250],{},[79,4710,98],{},[70,4712,1513],{},[79,4714,112],{"stretchy":91},[79,4716,4717],{},"≡",[79,4719,92],{"stretchy":91},[70,4721,4519],{},[79,4723,81],{},[70,4725,4637],{},[79,4727,98],{},[70,4729,1513],{},[79,4731,112],{"stretchy":91},[212,4733],{},[212,4735],{"width":4736},"1em",[79,4738,92],{"stretchy":91},[64,4740,4741,4743,4745],{},[70,4742,1250],{"mathvariant":115},[70,4744,1253],{"mathvariant":115},[70,4746,1256],{"mathvariant":115},[212,4748],{"width":4749},"0.3333em",[70,4751,4667],{},[79,4753,112],{"stretchy":91},[121,4755,4756],{"encoding":123},"(x + m*k) ≡ (y + n*k) \\pmod{L}",[42,4758,4760,4781,4799,4820,4841,4859,4878],{"className":4759,"ariaHidden":129},[128],[42,4761,4763,4766,4769,4772,4775,4778],{"className":4762},[133],[42,4764],{"className":4765,"style":227},[137],[42,4767,92],{"className":4768},[231],[42,4770,4489],{"className":4771},[142,146],[42,4773],{"className":4774,"style":238},[212],[42,4776,81],{"className":4777},[191],[42,4779],{"className":4780,"style":238},[212],[42,4782,4784,4787,4790,4793,4796],{"className":4783},[133],[42,4785],{"className":4786,"style":2661},[137],[42,4788,1250],{"className":4789},[142,146],[42,4791],{"className":4792,"style":238},[212],[42,4794,98],{"className":4795},[191],[42,4797],{"className":4798,"style":238},[212],[42,4800,4802,4805,4808,4811,4814,4817],{"className":4801},[133],[42,4803],{"className":4804,"style":227},[137],[42,4806,1513],{"className":4807,"style":1572},[142,146],[42,4809,112],{"className":4810},[316],[42,4812],{"className":4813,"style":213},[212],[42,4815,4717],{"className":4816},[217],[42,4818],{"className":4819,"style":213},[212],[42,4821,4823,4826,4829,4832,4835,4838],{"className":4822},[133],[42,4824],{"className":4825,"style":227},[137],[42,4827,92],{"className":4828},[231],[42,4830,4519],{"className":4831,"style":4535},[142,146],[42,4833],{"className":4834,"style":238},[212],[42,4836,81],{"className":4837},[191],[42,4839],{"className":4840,"style":238},[212],[42,4842,4844,4847,4850,4853,4856],{"className":4843},[133],[42,4845],{"className":4846,"style":2661},[137],[42,4848,4637],{"className":4849},[142,146],[42,4851],{"className":4852,"style":238},[212],[42,4854,98],{"className":4855},[191],[42,4857],{"className":4858,"style":238},[212],[42,4860,4862,4865,4868,4871,4874],{"className":4861},[133],[42,4863],{"className":4864,"style":227},[137],[42,4866,1513],{"className":4867,"style":1572},[142,146],[42,4869,112],{"className":4870},[316],[42,4872],{"className":4873},[212,1429],[42,4875],{"className":4876,"style":4877},[212],"margin-right:1em;",[42,4879,4881,4884,4887,4896,4900,4903],{"className":4880},[133],[42,4882],{"className":4883,"style":227},[137],[42,4885,92],{"className":4886},[231],[42,4888,4890],{"className":4889},[142],[42,4891,4893],{"className":4892},[142],[42,4894,1451],{"className":4895},[142,1450],[42,4897],{"className":4898,"style":4899},[212],"margin-right:0.3333em;",[42,4901,4667],{"className":4902},[142,146],[42,4904,112],{"className":4905},[316],[15,4907,4908,4909,4938,4939,413],{},"Where ",[42,4910,4912,4925],{"className":4911,"translate":46},[50],[42,4913,4915],{"className":4914},[54],[56,4916,4917],{"xmlns":58},[61,4918,4919,4923],{},[64,4920,4921],{},[79,4922,4717],{},[121,4924,4717],{"encoding":123},[42,4926,4928],{"className":4927,"ariaHidden":129},[128],[42,4929,4931,4935],{"className":4930},[133],[42,4932],{"className":4933,"style":4934},[137],"height:0.4637em;",[42,4936,4717],{"className":4937},[217]," denotes congruence modulo ",[42,4940,4942,4955],{"className":4941,"translate":46},[50],[42,4943,4945],{"className":4944},[54],[56,4946,4947],{"xmlns":58},[61,4948,4949,4953],{},[64,4950,4951],{},[70,4952,4667],{},[121,4954,4667],{"encoding":123},[42,4956,4958],{"className":4957,"ariaHidden":129},[128],[42,4959,4961,4964],{"className":4960},[133],[42,4962],{"className":4963,"style":351},[137],[42,4965,4667],{"className":4966},[142,146],[15,4968,4969],{},"Expanding this gives:",[42,4971,4973],{"className":4972,"translate":46},[45],[42,4974,4976,5024],{"className":4975,"translate":46},[50],[42,4977,4979],{"className":4978},[54],[56,4980,4981],{"xmlns":58,"display":59},[61,4982,4983,5021],{},[64,4984,4985,4987,4989,4991,4993,4995,4997,4999,5001,5003,5005,5007,5009,5011,5013],{},[70,4986,4489],{},[79,4988,81],{},[70,4990,1250],{},[79,4992,98],{},[70,4994,1513],{},[79,4996,88],{},[70,4998,4519],{},[79,5000,81],{},[70,5002,4637],{},[79,5004,98],{},[70,5006,1513],{},[79,5008,81],{},[70,5010,4667],{},[79,5012,98],{},[1070,5014,5015,5017],{},[70,5016,1513],{},[79,5018,5020],{"mathvariant":115,"lspace":5019,"rspace":5019},"0em","′",[121,5022,5023],{"encoding":123},"x + m*k = y + n*k + L*k'",[42,5025,5027,5046,5064,5082,5101,5119,5138,5156],{"className":5026,"ariaHidden":129},[128],[42,5028,5030,5034,5037,5040,5043],{"className":5029},[133],[42,5031],{"className":5032,"style":5033},[137],"height:0.6667em;vertical-align:-0.0833em;",[42,5035,4489],{"className":5036},[142,146],[42,5038],{"className":5039,"style":238},[212],[42,5041,81],{"className":5042},[191],[42,5044],{"className":5045,"style":238},[212],[42,5047,5049,5052,5055,5058,5061],{"className":5048},[133],[42,5050],{"className":5051,"style":2661},[137],[42,5053,1250],{"className":5054},[142,146],[42,5056],{"className":5057,"style":238},[212],[42,5059,98],{"className":5060},[191],[42,5062],{"className":5063,"style":238},[212],[42,5065,5067,5070,5073,5076,5079],{"className":5066},[133],[42,5068],{"className":5069,"style":1440},[137],[42,5071,1513],{"className":5072,"style":1572},[142,146],[42,5074],{"className":5075,"style":213},[212],[42,5077,88],{"className":5078},[217],[42,5080],{"className":5081,"style":213},[212],[42,5083,5085,5089,5092,5095,5098],{"className":5084},[133],[42,5086],{"className":5087,"style":5088},[137],"height:0.7778em;vertical-align:-0.1944em;",[42,5090,4519],{"className":5091,"style":4535},[142,146],[42,5093],{"className":5094,"style":238},[212],[42,5096,81],{"className":5097},[191],[42,5099],{"className":5100,"style":238},[212],[42,5102,5104,5107,5110,5113,5116],{"className":5103},[133],[42,5105],{"className":5106,"style":2661},[137],[42,5108,4637],{"className":5109},[142,146],[42,5111],{"className":5112,"style":238},[212],[42,5114,98],{"className":5115},[191],[42,5117],{"className":5118,"style":238},[212],[42,5120,5122,5126,5129,5132,5135],{"className":5121},[133],[42,5123],{"className":5124,"style":5125},[137],"height:0.7778em;vertical-align:-0.0833em;",[42,5127,1513],{"className":5128,"style":1572},[142,146],[42,5130],{"className":5131,"style":238},[212],[42,5133,81],{"className":5134},[191],[42,5136],{"className":5137,"style":238},[212],[42,5139,5141,5144,5147,5150,5153],{"className":5140},[133],[42,5142],{"className":5143,"style":351},[137],[42,5145,4667],{"className":5146},[142,146],[42,5148],{"className":5149,"style":238},[212],[42,5151,98],{"className":5152},[191],[42,5154],{"className":5155,"style":238},[212],[42,5157,5159,5163],{"className":5158},[133],[42,5160],{"className":5161,"style":5162},[137],"height:0.8019em;",[42,5164,5166,5169],{"className":5165},[142],[42,5167,1513],{"className":5168,"style":1572},[142,146],[42,5170,5172],{"className":5171},[151],[42,5173,5175],{"className":5174},[155],[42,5176,5178],{"className":5177},[160],[42,5179,5181],{"className":5180,"style":5162},[164],[42,5182,5184,5187],{"style":5183},"top:-3.113em;margin-right:0.05em;",[42,5185],{"className":5186,"style":173},[172],[42,5188,5190],{"className":5189},[177,178,179,180],[42,5191,5193],{"className":5192},[142,180],[42,5194,5020],{"className":5195},[142,180],[15,5197,5198],{},"Rearranging terms, we get:",[42,5200,5202],{"className":5201,"translate":46},[45],[42,5203,5205,5251],{"className":5204,"translate":46},[50],[42,5206,5208],{"className":5207},[54],[56,5209,5210],{"xmlns":58,"display":59},[61,5211,5212,5248],{},[64,5213,5214,5216,5218,5220,5222,5224,5226,5228,5230,5232,5234,5236,5238,5240,5242],{},[70,5215,4489],{},[79,5217,674],{},[70,5219,4519],{},[79,5221,88],{},[79,5223,92],{"stretchy":91},[70,5225,4637],{},[79,5227,674],{},[70,5229,1250],{},[79,5231,112],{"stretchy":91},[79,5233,98],{},[70,5235,1513],{},[79,5237,81],{},[70,5239,4667],{},[79,5241,98],{},[1070,5243,5244,5246],{},[70,5245,1513],{},[79,5247,5020],{"mathvariant":115,"lspace":5019,"rspace":5019},[121,5249,5250],{"encoding":123},"x - y = (n - m)*k + L*k'",[42,5252,5254,5272,5290,5311,5332,5350,5368],{"className":5253,"ariaHidden":129},[128],[42,5255,5257,5260,5263,5266,5269],{"className":5256},[133],[42,5258],{"className":5259,"style":5033},[137],[42,5261,4489],{"className":5262},[142,146],[42,5264],{"className":5265,"style":238},[212],[42,5267,674],{"className":5268},[191],[42,5270],{"className":5271,"style":238},[212],[42,5273,5275,5278,5281,5284,5287],{"className":5274},[133],[42,5276],{"className":5277,"style":4531},[137],[42,5279,4519],{"className":5280,"style":4535},[142,146],[42,5282],{"className":5283,"style":213},[212],[42,5285,88],{"className":5286},[217],[42,5288],{"className":5289,"style":213},[212],[42,5291,5293,5296,5299,5302,5305,5308],{"className":5292},[133],[42,5294],{"className":5295,"style":227},[137],[42,5297,92],{"className":5298},[231],[42,5300,4637],{"className":5301},[142,146],[42,5303],{"className":5304,"style":238},[212],[42,5306,674],{"className":5307},[191],[42,5309],{"className":5310,"style":238},[212],[42,5312,5314,5317,5320,5323,5326,5329],{"className":5313},[133],[42,5315],{"className":5316,"style":227},[137],[42,5318,1250],{"className":5319},[142,146],[42,5321,112],{"className":5322},[316],[42,5324],{"className":5325,"style":238},[212],[42,5327,98],{"className":5328},[191],[42,5330],{"className":5331,"style":238},[212],[42,5333,5335,5338,5341,5344,5347],{"className":5334},[133],[42,5336],{"className":5337,"style":5125},[137],[42,5339,1513],{"className":5340,"style":1572},[142,146],[42,5342],{"className":5343,"style":238},[212],[42,5345,81],{"className":5346},[191],[42,5348],{"className":5349,"style":238},[212],[42,5351,5353,5356,5359,5362,5365],{"className":5352},[133],[42,5354],{"className":5355,"style":351},[137],[42,5357,4667],{"className":5358},[142,146],[42,5360],{"className":5361,"style":238},[212],[42,5363,98],{"className":5364},[191],[42,5366],{"className":5367,"style":238},[212],[42,5369,5371,5374],{"className":5370},[133],[42,5372],{"className":5373,"style":5162},[137],[42,5375,5377,5380],{"className":5376},[142],[42,5378,1513],{"className":5379,"style":1572},[142,146],[42,5381,5383],{"className":5382},[151],[42,5384,5386],{"className":5385},[155],[42,5387,5389],{"className":5388},[160],[42,5390,5392],{"className":5391,"style":5162},[164],[42,5393,5394,5397],{"style":5183},[42,5395],{"className":5396,"style":173},[172],[42,5398,5400],{"className":5399},[177,178,179,180],[42,5401,5403],{"className":5402},[142,180],[42,5404,5020],{"className":5405},[142,180],[15,5407,5408],{},"Thus,",[42,5410,5412],{"className":5411,"translate":46},[45],[42,5413,5415,5473],{"className":5414,"translate":46},[50],[42,5416,5418],{"className":5417},[54],[56,5419,5420],{"xmlns":58,"display":59},[61,5421,5422,5470],{},[64,5423,5424,5426,5428,5430,5432,5434,5436,5438,5440,5442,5444,5446,5448,5450,5452,5454,5456,5464,5466,5468],{},[79,5425,92],{"stretchy":91},[70,5427,4489],{},[79,5429,674],{},[70,5431,4519],{},[79,5433,112],{"stretchy":91},[79,5435,4717],{},[79,5437,92],{"stretchy":91},[70,5439,4637],{},[79,5441,674],{},[70,5443,1250],{},[79,5445,112],{"stretchy":91},[79,5447,98],{},[70,5449,1513],{},[212,5451],{},[212,5453],{"width":4736},[79,5455,92],{"stretchy":91},[64,5457,5458,5460,5462],{},[70,5459,1250],{"mathvariant":115},[70,5461,1253],{"mathvariant":115},[70,5463,1256],{"mathvariant":115},[212,5465],{"width":4749},[70,5467,4667],{},[79,5469,112],{"stretchy":91},[121,5471,5472],{"encoding":123},"(x - y) ≡ (n - m)*k \\pmod{L}",[42,5474,5476,5497,5518,5539,5560,5575],{"className":5475,"ariaHidden":129},[128],[42,5477,5479,5482,5485,5488,5491,5494],{"className":5478},[133],[42,5480],{"className":5481,"style":227},[137],[42,5483,92],{"className":5484},[231],[42,5486,4489],{"className":5487},[142,146],[42,5489],{"className":5490,"style":238},[212],[42,5492,674],{"className":5493},[191],[42,5495],{"className":5496,"style":238},[212],[42,5498,5500,5503,5506,5509,5512,5515],{"className":5499},[133],[42,5501],{"className":5502,"style":227},[137],[42,5504,4519],{"className":5505,"style":4535},[142,146],[42,5507,112],{"className":5508},[316],[42,5510],{"className":5511,"style":213},[212],[42,5513,4717],{"className":5514},[217],[42,5516],{"className":5517,"style":213},[212],[42,5519,5521,5524,5527,5530,5533,5536],{"className":5520},[133],[42,5522],{"className":5523,"style":227},[137],[42,5525,92],{"className":5526},[231],[42,5528,4637],{"className":5529},[142,146],[42,5531],{"className":5532,"style":238},[212],[42,5534,674],{"className":5535},[191],[42,5537],{"className":5538,"style":238},[212],[42,5540,5542,5545,5548,5551,5554,5557],{"className":5541},[133],[42,5543],{"className":5544,"style":227},[137],[42,5546,1250],{"className":5547},[142,146],[42,5549,112],{"className":5550},[316],[42,5552],{"className":5553,"style":238},[212],[42,5555,98],{"className":5556},[191],[42,5558],{"className":5559,"style":238},[212],[42,5561,5563,5566,5569,5572],{"className":5562},[133],[42,5564],{"className":5565,"style":1440},[137],[42,5567,1513],{"className":5568,"style":1572},[142,146],[42,5570],{"className":5571},[212,1429],[42,5573],{"className":5574,"style":4877},[212],[42,5576,5578,5581,5584,5593,5596,5599],{"className":5577},[133],[42,5579],{"className":5580,"style":227},[137],[42,5582,92],{"className":5583},[231],[42,5585,5587],{"className":5586},[142],[42,5588,5590],{"className":5589},[142],[42,5591,1451],{"className":5592},[142,1450],[42,5594],{"className":5595,"style":4899},[212],[42,5597,4667],{"className":5598},[142,146],[42,5600,112],{"className":5601},[316],[15,5603,5604,5605,596,5678,445,5752,5755],{},"Let ",[42,5606,5608,5630],{"className":5607,"translate":46},[50],[42,5609,5611],{"className":5610},[54],[56,5612,5613],{"xmlns":58},[61,5614,5615,5627],{},[64,5616,5617,5619,5621,5623,5625],{},[70,5618,4637],{},[79,5620,674],{},[70,5622,1250],{},[79,5624,88],{},[70,5626,2626],{},[121,5628,5629],{"encoding":123},"n - m = a",[42,5631,5633,5651,5669],{"className":5632,"ariaHidden":129},[128],[42,5634,5636,5639,5642,5645,5648],{"className":5635},[133],[42,5637],{"className":5638,"style":5033},[137],[42,5640,4637],{"className":5641},[142,146],[42,5643],{"className":5644,"style":238},[212],[42,5646,674],{"className":5647},[191],[42,5649],{"className":5650,"style":238},[212],[42,5652,5654,5657,5660,5663,5666],{"className":5653},[133],[42,5655],{"className":5656,"style":4501},[137],[42,5658,1250],{"className":5659},[142,146],[42,5661],{"className":5662,"style":213},[212],[42,5664,88],{"className":5665},[217],[42,5667],{"className":5668,"style":213},[212],[42,5670,5672,5675],{"className":5671},[133],[42,5673],{"className":5674,"style":4501},[137],[42,5676,2626],{"className":5677},[142,146],[42,5679,5681,5704],{"className":5680,"translate":46},[50],[42,5682,5684],{"className":5683},[54],[56,5685,5686],{"xmlns":58},[61,5687,5688,5701],{},[64,5689,5690,5692,5694,5696,5698],{},[70,5691,4489],{},[79,5693,674],{},[70,5695,4519],{},[79,5697,88],{},[70,5699,5700],{},"b",[121,5702,5703],{"encoding":123},"x - y = b",[42,5705,5707,5725,5743],{"className":5706,"ariaHidden":129},[128],[42,5708,5710,5713,5716,5719,5722],{"className":5709},[133],[42,5711],{"className":5712,"style":5033},[137],[42,5714,4489],{"className":5715},[142,146],[42,5717],{"className":5718,"style":238},[212],[42,5720,674],{"className":5721},[191],[42,5723],{"className":5724,"style":238},[212],[42,5726,5728,5731,5734,5737,5740],{"className":5727},[133],[42,5729],{"className":5730,"style":4531},[137],[42,5732,4519],{"className":5733,"style":4535},[142,146],[42,5735],{"className":5736,"style":213},[212],[42,5738,88],{"className":5739},[217],[42,5741],{"className":5742,"style":213},[212],[42,5744,5746,5749],{"className":5745},[133],[42,5747],{"className":5748,"style":1440},[137],[42,5750,5700],{"className":5751},[142,146],[5753,5754],"br",{},"\nyielding the standard linear congruence:",[42,5757,5759],{"className":5758,"translate":46},[45],[42,5760,5762,5804],{"className":5761,"translate":46},[50],[42,5763,5765],{"className":5764},[54],[56,5766,5767],{"xmlns":58,"display":59},[61,5768,5769,5801],{},[64,5770,5771,5773,5775,5777,5779,5781,5783,5785,5787,5795,5797,5799],{},[70,5772,2626],{},[79,5774,98],{},[70,5776,1513],{},[79,5778,4717],{},[70,5780,5700],{},[212,5782],{},[212,5784],{"width":4736},[79,5786,92],{"stretchy":91},[64,5788,5789,5791,5793],{},[70,5790,1250],{"mathvariant":115},[70,5792,1253],{"mathvariant":115},[70,5794,1256],{"mathvariant":115},[212,5796],{"width":4749},[70,5798,4667],{},[79,5800,112],{"stretchy":91},[121,5802,5803],{"encoding":123},"a*k ≡ b \\pmod{L}",[42,5805,5807,5825,5843,5858],{"className":5806,"ariaHidden":129},[128],[42,5808,5810,5813,5816,5819,5822],{"className":5809},[133],[42,5811],{"className":5812,"style":2661},[137],[42,5814,2626],{"className":5815},[142,146],[42,5817],{"className":5818,"style":238},[212],[42,5820,98],{"className":5821},[191],[42,5823],{"className":5824,"style":238},[212],[42,5826,5828,5831,5834,5837,5840],{"className":5827},[133],[42,5829],{"className":5830,"style":1440},[137],[42,5832,1513],{"className":5833,"style":1572},[142,146],[42,5835],{"className":5836,"style":213},[212],[42,5838,4717],{"className":5839},[217],[42,5841],{"className":5842,"style":213},[212],[42,5844,5846,5849,5852,5855],{"className":5845},[133],[42,5847],{"className":5848,"style":1440},[137],[42,5850,5700],{"className":5851},[142,146],[42,5853],{"className":5854},[212,1429],[42,5856],{"className":5857,"style":4877},[212],[42,5859,5861,5864,5867,5876,5879,5882],{"className":5860},[133],[42,5862],{"className":5863,"style":227},[137],[42,5865,92],{"className":5866},[231],[42,5868,5870],{"className":5869},[142],[42,5871,5873],{"className":5872},[142],[42,5874,1451],{"className":5875},[142,1450],[42,5877],{"className":5878,"style":4899},[212],[42,5880,4667],{"className":5881},[142,146],[42,5883,112],{"className":5884},[316],[19,5886,5887],{},[15,5888,5889,5890,596,5918,5946,5947,6075,6076,6104,6105,596,6133,6218,6219,413],{},"Definition: If ",[42,5891,5893,5906],{"className":5892,"translate":46},[50],[42,5894,5896],{"className":5895},[54],[56,5897,5898],{"xmlns":58},[61,5899,5900,5904],{},[64,5901,5902],{},[70,5903,2626],{},[121,5905,2626],{"encoding":123},[42,5907,5909],{"className":5908,"ariaHidden":129},[128],[42,5910,5912,5915],{"className":5911},[133],[42,5913],{"className":5914,"style":4501},[137],[42,5916,2626],{"className":5917},[142,146],[42,5919,5921,5934],{"className":5920,"translate":46},[50],[42,5922,5924],{"className":5923},[54],[56,5925,5926],{"xmlns":58},[61,5927,5928,5932],{},[64,5929,5930],{},[70,5931,5700],{},[121,5933,5700],{"encoding":123},[42,5935,5937],{"className":5936,"ariaHidden":129},[128],[42,5938,5940,5943],{"className":5939},[133],[42,5941],{"className":5942,"style":1440},[137],[42,5944,5700],{"className":5945},[142,146]," are integers, the expression ",[42,5948,5950,5993],{"className":5949,"translate":46},[50],[42,5951,5953],{"className":5952},[54],[56,5954,5955],{"xmlns":58},[61,5956,5957,5990],{},[64,5958,5959,5961,5963,5965,5967,5969,5971,5974,5976,5984,5986,5988],{},[70,5960,2626],{},[79,5962,98],{},[70,5964,4489],{},[79,5966,4717],{},[70,5968,5700],{},[212,5970],{},[212,5972],{"width":5973},"0.4444em",[79,5975,92],{"stretchy":91},[64,5977,5978,5980,5982],{},[70,5979,1250],{"mathvariant":115},[70,5981,1253],{"mathvariant":115},[70,5983,1256],{"mathvariant":115},[212,5985],{"width":4749},[70,5987,119],{},[79,5989,112],{"stretchy":91},[121,5991,5992],{"encoding":123},"a*x ≡ b \\pmod{M}",[42,5994,5996,6014,6032,6048],{"className":5995,"ariaHidden":129},[128],[42,5997,5999,6002,6005,6008,6011],{"className":5998},[133],[42,6000],{"className":6001,"style":2661},[137],[42,6003,2626],{"className":6004},[142,146],[42,6006],{"className":6007,"style":238},[212],[42,6009,98],{"className":6010},[191],[42,6012],{"className":6013,"style":238},[212],[42,6015,6017,6020,6023,6026,6029],{"className":6016},[133],[42,6018],{"className":6019,"style":4934},[137],[42,6021,4489],{"className":6022},[142,146],[42,6024],{"className":6025,"style":213},[212],[42,6027,4717],{"className":6028},[217],[42,6030],{"className":6031,"style":213},[212],[42,6033,6035,6038,6041,6044],{"className":6034},[133],[42,6036],{"className":6037,"style":1440},[137],[42,6039,5700],{"className":6040},[142,146],[42,6042],{"className":6043},[212,1429],[42,6045],{"className":6046,"style":6047},[212],"margin-right:0.4444em;",[42,6049,6051,6054,6057,6066,6069,6072],{"className":6050},[133],[42,6052],{"className":6053,"style":227},[137],[42,6055,92],{"className":6056},[231],[42,6058,6060],{"className":6059},[142],[42,6061,6063],{"className":6062},[142],[42,6064,1451],{"className":6065},[142,1450],[42,6067],{"className":6068,"style":4899},[212],[42,6070,119],{"className":6071,"style":147},[142,146],[42,6073,112],{"className":6074},[316],", where ",[42,6077,6079,6092],{"className":6078,"translate":46},[50],[42,6080,6082],{"className":6081},[54],[56,6083,6084],{"xmlns":58},[61,6085,6086,6090],{},[64,6087,6088],{},[70,6089,4489],{},[121,6091,4489],{"encoding":123},[42,6093,6095],{"className":6094,"ariaHidden":129},[128],[42,6096,6098,6101],{"className":6097},[133],[42,6099],{"className":6100,"style":4501},[137],[42,6102,4489],{"className":6103},[142,146]," is an unknown integer, is called a linear congruence. Here, ",[42,6106,6108,6121],{"className":6107,"translate":46},[50],[42,6109,6111],{"className":6110},[54],[56,6112,6113],{"xmlns":58},[61,6114,6115,6119],{},[64,6116,6117],{},[79,6118,4717],{},[121,6120,4717],{"encoding":123},[42,6122,6124],{"className":6123,"ariaHidden":129},[128],[42,6125,6127,6130],{"className":6126},[133],[42,6128],{"className":6129,"style":4934},[137],[42,6131,4717],{"className":6132},[217],[42,6134,6136,6172],{"className":6135,"translate":46},[50],[42,6137,6139],{"className":6138},[54],[56,6140,6141],{"xmlns":58},[61,6142,6143,6169],{},[64,6144,6145,6147,6149,6151,6153,6161,6163,6165,6167],{},[79,6146,92],{"stretchy":91},[212,6148],{},[212,6150],{"width":5973},[79,6152,92],{"stretchy":91},[64,6154,6155,6157,6159],{},[70,6156,1250],{"mathvariant":115},[70,6158,1253],{"mathvariant":115},[70,6160,1256],{"mathvariant":115},[212,6162],{"width":4749},[70,6164,119],{},[79,6166,112],{"stretchy":91},[79,6168,112],{"stretchy":91},[121,6170,6171],{"encoding":123},"(\\pmod{M})",[42,6173,6175,6190],{"className":6174,"ariaHidden":129},[128],[42,6176,6178,6181,6184,6187],{"className":6177},[133],[42,6179],{"className":6180,"style":227},[137],[42,6182,92],{"className":6183},[231],[42,6185],{"className":6186},[212,1429],[42,6188],{"className":6189,"style":6047},[212],[42,6191,6193,6196,6199,6208,6211,6214],{"className":6192},[133],[42,6194],{"className":6195,"style":227},[137],[42,6197,92],{"className":6198},[231],[42,6200,6202],{"className":6201},[142],[42,6203,6205],{"className":6204},[142],[42,6206,1451],{"className":6207},[142,1450],[42,6209],{"className":6210,"style":4899},[212],[42,6212,119],{"className":6213,"style":147},[142,146],[42,6215,6217],{"className":6216},[316],"))"," indicate that both sides are taken modulo ",[42,6220,6222,6235],{"className":6221,"translate":46},[50],[42,6223,6225],{"className":6224},[54],[56,6226,6227],{"xmlns":58},[61,6228,6229,6233],{},[64,6230,6231],{},[70,6232,119],{},[121,6234,119],{"encoding":123},[42,6236,6238],{"className":6237,"ariaHidden":129},[128],[42,6239,6241,6244],{"className":6240},[133],[42,6242],{"className":6243,"style":351},[137],[42,6245,119],{"className":6246,"style":147},[142,146],[15,6248,6249],{},"To solve a linear congruence, we proceed with the following transformations:",[6251,6252,6253,6499,7636],"ol",{},[566,6254,6255,6256,6381,6382,413],{},"The standard linear congruence ",[42,6257,6259,6300],{"className":6258,"translate":46},[50],[42,6260,6262],{"className":6261},[54],[56,6263,6264],{"xmlns":58},[61,6265,6266,6298],{},[64,6267,6268,6270,6272,6274,6276,6278,6280,6282,6284,6292,6294,6296],{},[70,6269,2626],{},[79,6271,98],{},[70,6273,4489],{},[79,6275,4717],{},[70,6277,5700],{},[212,6279],{},[212,6281],{"width":5973},[79,6283,92],{"stretchy":91},[64,6285,6286,6288,6290],{},[70,6287,1250],{"mathvariant":115},[70,6289,1253],{"mathvariant":115},[70,6291,1256],{"mathvariant":115},[212,6293],{"width":4749},[70,6295,119],{},[79,6297,112],{"stretchy":91},[121,6299,5992],{"encoding":123},[42,6301,6303,6321,6339,6354],{"className":6302,"ariaHidden":129},[128],[42,6304,6306,6309,6312,6315,6318],{"className":6305},[133],[42,6307],{"className":6308,"style":2661},[137],[42,6310,2626],{"className":6311},[142,146],[42,6313],{"className":6314,"style":238},[212],[42,6316,98],{"className":6317},[191],[42,6319],{"className":6320,"style":238},[212],[42,6322,6324,6327,6330,6333,6336],{"className":6323},[133],[42,6325],{"className":6326,"style":4934},[137],[42,6328,4489],{"className":6329},[142,146],[42,6331],{"className":6332,"style":213},[212],[42,6334,4717],{"className":6335},[217],[42,6337],{"className":6338,"style":213},[212],[42,6340,6342,6345,6348,6351],{"className":6341},[133],[42,6343],{"className":6344,"style":1440},[137],[42,6346,5700],{"className":6347},[142,146],[42,6349],{"className":6350},[212,1429],[42,6352],{"className":6353,"style":6047},[212],[42,6355,6357,6360,6363,6372,6375,6378],{"className":6356},[133],[42,6358],{"className":6359,"style":227},[137],[42,6361,92],{"className":6362},[231],[42,6364,6366],{"className":6365},[142],[42,6367,6369],{"className":6368},[142],[42,6370,1451],{"className":6371},[142,1450],[42,6373],{"className":6374,"style":4899},[212],[42,6376,119],{"className":6377,"style":147},[142,146],[42,6379,112],{"className":6380},[316]," is equivalent to the Diophantine equation ",[42,6383,6385,6415],{"className":6384,"translate":46},[50],[42,6386,6388],{"className":6387},[54],[56,6389,6390],{"xmlns":58},[61,6391,6392,6412],{},[64,6393,6394,6396,6398,6400,6402,6404,6406,6408,6410],{},[70,6395,2626],{},[79,6397,98],{},[70,6399,4489],{},[79,6401,81],{},[70,6403,119],{},[79,6405,98],{},[70,6407,4519],{},[79,6409,88],{},[70,6411,5700],{},[121,6413,6414],{"encoding":123},"a*x + M*y = b",[42,6416,6418,6436,6454,6472,6490],{"className":6417,"ariaHidden":129},[128],[42,6419,6421,6424,6427,6430,6433],{"className":6420},[133],[42,6422],{"className":6423,"style":2661},[137],[42,6425,2626],{"className":6426},[142,146],[42,6428],{"className":6429,"style":238},[212],[42,6431,98],{"className":6432},[191],[42,6434],{"className":6435,"style":238},[212],[42,6437,6439,6442,6445,6448,6451],{"className":6438},[133],[42,6440],{"className":6441,"style":5033},[137],[42,6443,4489],{"className":6444},[142,146],[42,6446],{"className":6447,"style":238},[212],[42,6449,81],{"className":6450},[191],[42,6452],{"className":6453,"style":238},[212],[42,6455,6457,6460,6463,6466,6469],{"className":6456},[133],[42,6458],{"className":6459,"style":351},[137],[42,6461,119],{"className":6462,"style":147},[142,146],[42,6464],{"className":6465,"style":238},[212],[42,6467,98],{"className":6468},[191],[42,6470],{"className":6471,"style":238},[212],[42,6473,6475,6478,6481,6484,6487],{"className":6474},[133],[42,6476],{"className":6477,"style":4531},[137],[42,6479,4519],{"className":6480,"style":4535},[142,146],[42,6482],{"className":6483,"style":213},[212],[42,6485,88],{"className":6486},[217],[42,6488],{"className":6489,"style":213},[212],[42,6491,6493,6496],{"className":6492},[133],[42,6494],{"className":6495,"style":1440},[137],[42,6497,5700],{"className":6498},[142,146],[566,6500,5604,6501,6529,6530,596,6558,6586,6587,6678,6679,6707,6708,6736,6737,6075,6980,445,7094,474,7208,7322,7323,596,7394,7464,7465,413],{},[42,6502,6504,6517],{"className":6503,"translate":46},[50],[42,6505,6507],{"className":6506},[54],[56,6508,6509],{"xmlns":58},[61,6510,6511,6515],{},[64,6512,6513],{},[70,6514,1256],{},[121,6516,1256],{"encoding":123},[42,6518,6520],{"className":6519,"ariaHidden":129},[128],[42,6521,6523,6526],{"className":6522},[133],[42,6524],{"className":6525,"style":1440},[137],[42,6527,1256],{"className":6528},[142,146]," be the greatest common divisor (gcd) of ",[42,6531,6533,6546],{"className":6532,"translate":46},[50],[42,6534,6536],{"className":6535},[54],[56,6537,6538],{"xmlns":58},[61,6539,6540,6544],{},[64,6541,6542],{},[70,6543,2626],{},[121,6545,2626],{"encoding":123},[42,6547,6549],{"className":6548,"ariaHidden":129},[128],[42,6550,6552,6555],{"className":6551},[133],[42,6553],{"className":6554,"style":4501},[137],[42,6556,2626],{"className":6557},[142,146],[42,6559,6561,6574],{"className":6560,"translate":46},[50],[42,6562,6564],{"className":6563},[54],[56,6565,6566],{"xmlns":58},[61,6567,6568,6572],{},[64,6569,6570],{},[70,6571,119],{},[121,6573,119],{"encoding":123},[42,6575,6577],{"className":6576,"ariaHidden":129},[128],[42,6578,6580,6583],{"className":6579},[133],[42,6581],{"className":6582,"style":351},[137],[42,6584,119],{"className":6585,"style":147},[142,146],", denoted ",[42,6588,6590,6624],{"className":6589,"translate":46},[50],[42,6591,6593],{"className":6592},[54],[56,6594,6595],{"xmlns":58},[61,6596,6597,6621],{},[64,6598,6599,6602,6605,6607,6609,6611,6613,6615,6617,6619],{},[70,6600,6601],{},"g",[70,6603,6604],{},"c",[70,6606,1256],{},[79,6608,92],{"stretchy":91},[70,6610,2626],{},[79,6612,3985],{"separator":129},[70,6614,119],{},[79,6616,112],{"stretchy":91},[79,6618,88],{},[70,6620,1256],{},[121,6622,6623],{"encoding":123},"gcd(a,M) = d",[42,6625,6627,6669],{"className":6626,"ariaHidden":129},[128],[42,6628,6630,6633,6636,6639,6642,6645,6648,6651,6654,6657,6660,6663,6666],{"className":6629},[133],[42,6631],{"className":6632,"style":227},[137],[42,6634,6601],{"className":6635,"style":4535},[142,146],[42,6637,6604],{"className":6638},[142,146],[42,6640,1256],{"className":6641},[142,146],[42,6643,92],{"className":6644},[231],[42,6646,2626],{"className":6647},[142,146],[42,6649,3985],{"className":6650},[4035],[42,6652],{"className":6653,"style":1455},[212],[42,6655,119],{"className":6656,"style":147},[142,146],[42,6658,112],{"className":6659},[316],[42,6661],{"className":6662,"style":213},[212],[42,6664,88],{"className":6665},[217],[42,6667],{"className":6668,"style":213},[212],[42,6670,6672,6675],{"className":6671},[133],[42,6673],{"className":6674,"style":1440},[137],[42,6676,1256],{"className":6677},[142,146],". If the equation has a solution, ",[42,6680,6682,6695],{"className":6681,"translate":46},[50],[42,6683,6685],{"className":6684},[54],[56,6686,6687],{"xmlns":58},[61,6688,6689,6693],{},[64,6690,6691],{},[70,6692,1256],{},[121,6694,1256],{"encoding":123},[42,6696,6698],{"className":6697,"ariaHidden":129},[128],[42,6699,6701,6704],{"className":6700},[133],[42,6702],{"className":6703,"style":1440},[137],[42,6705,1256],{"className":6706},[142,146]," must divide ",[42,6709,6711,6724],{"className":6710,"translate":46},[50],[42,6712,6714],{"className":6713},[54],[56,6715,6716],{"xmlns":58},[61,6717,6718,6722],{},[64,6719,6720],{},[70,6721,5700],{},[121,6723,5700],{"encoding":123},[42,6725,6727],{"className":6726,"ariaHidden":129},[128],[42,6728,6730,6733],{"className":6729},[133],[42,6731],{"className":6732,"style":1440},[137],[42,6734,5700],{"className":6735},[142,146],". We rewrite the equation as ",[42,6738,6740,6782],{"className":6739,"translate":46},[50],[42,6741,6743],{"className":6742},[54],[56,6744,6745],{"xmlns":58},[61,6746,6747,6779],{},[64,6748,6749,6755,6757,6759,6761,6767,6769,6771,6773],{},[67,6750,6751,6753],{},[70,6752,2626],{},[83,6754,739],{},[79,6756,98],{},[70,6758,4489],{},[79,6760,81],{},[67,6762,6763,6765],{},[70,6764,119],{},[83,6766,739],{},[79,6768,98],{},[70,6770,4519],{},[79,6772,88],{},[67,6774,6775,6777],{},[70,6776,5700],{},[83,6778,739],{},[121,6780,6781],{"encoding":123},"a_0*x + M_0*y = b_0",[42,6783,6785,6842,6860,6915,6933],{"className":6784,"ariaHidden":129},[128],[42,6786,6788,6792,6833,6836,6839],{"className":6787},[133],[42,6789],{"className":6790,"style":6791},[137],"height:0.6153em;vertical-align:-0.15em;",[42,6793,6795,6798],{"className":6794},[142],[42,6796,2626],{"className":6797},[142,146],[42,6799,6801],{"className":6800},[151],[42,6802,6804,6825],{"className":6803},[155,156],[42,6805,6807,6822],{"className":6806},[160],[42,6808,6810],{"className":6809,"style":937},[164],[42,6811,6813,6816],{"style":6812},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[42,6814],{"className":6815,"style":173},[172],[42,6817,6819],{"className":6818},[177,178,179,180],[42,6820,739],{"className":6821},[142,180],[42,6823,199],{"className":6824},[198],[42,6826,6828],{"className":6827},[160],[42,6829,6831],{"className":6830,"style":961},[164],[42,6832],{},[42,6834],{"className":6835,"style":238},[212],[42,6837,98],{"className":6838},[191],[42,6840],{"className":6841,"style":238},[212],[42,6843,6845,6848,6851,6854,6857],{"className":6844},[133],[42,6846],{"className":6847,"style":5033},[137],[42,6849,4489],{"className":6850},[142,146],[42,6852],{"className":6853,"style":238},[212],[42,6855,81],{"className":6856},[191],[42,6858],{"className":6859,"style":238},[212],[42,6861,6863,6866,6906,6909,6912],{"className":6862},[133],[42,6864],{"className":6865,"style":918},[137],[42,6867,6869,6872],{"className":6868},[142],[42,6870,119],{"className":6871,"style":147},[142,146],[42,6873,6875],{"className":6874},[151],[42,6876,6878,6898],{"className":6877},[155,156],[42,6879,6881,6895],{"className":6880},[160],[42,6882,6884],{"className":6883,"style":937},[164],[42,6885,6886,6889],{"style":168},[42,6887],{"className":6888,"style":173},[172],[42,6890,6892],{"className":6891},[177,178,179,180],[42,6893,739],{"className":6894},[142,180],[42,6896,199],{"className":6897},[198],[42,6899,6901],{"className":6900},[160],[42,6902,6904],{"className":6903,"style":961},[164],[42,6905],{},[42,6907],{"className":6908,"style":238},[212],[42,6910,98],{"className":6911},[191],[42,6913],{"className":6914,"style":238},[212],[42,6916,6918,6921,6924,6927,6930],{"className":6917},[133],[42,6919],{"className":6920,"style":4531},[137],[42,6922,4519],{"className":6923,"style":4535},[142,146],[42,6925],{"className":6926,"style":213},[212],[42,6928,88],{"className":6929},[217],[42,6931],{"className":6932,"style":213},[212],[42,6934,6936,6940],{"className":6935},[133],[42,6937],{"className":6938,"style":6939},[137],"height:0.8444em;vertical-align:-0.15em;",[42,6941,6943,6946],{"className":6942},[142],[42,6944,5700],{"className":6945},[142,146],[42,6947,6949],{"className":6948},[151],[42,6950,6952,6972],{"className":6951},[155,156],[42,6953,6955,6969],{"className":6954},[160],[42,6956,6958],{"className":6957,"style":937},[164],[42,6959,6960,6963],{"style":6812},[42,6961],{"className":6962,"style":173},[172],[42,6964,6966],{"className":6965},[177,178,179,180],[42,6967,739],{"className":6968},[142,180],[42,6970,199],{"className":6971},[198],[42,6973,6975],{"className":6974},[160],[42,6976,6978],{"className":6977,"style":961},[164],[42,6979],{},[42,6981,6983,7009],{"className":6982,"translate":46},[50],[42,6984,6986],{"className":6985},[54],[56,6987,6988],{"xmlns":58},[61,6989,6990,7006],{},[64,6991,6992,6994,6996,7002,7004],{},[70,6993,2626],{},[79,6995,88],{},[67,6997,6998,7000],{},[70,6999,2626],{},[83,7001,739],{},[79,7003,98],{},[70,7005,1256],{},[121,7007,7008],{"encoding":123},"a = 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= 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= 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Here, ",[42,7324,7326,7344],{"className":7325,"translate":46},[50],[42,7327,7329],{"className":7328},[54],[56,7330,7331],{"xmlns":58},[61,7332,7333,7341],{},[64,7334,7335],{},[67,7336,7337,7339],{},[70,7338,2626],{},[83,7340,739],{},[121,7342,7343],{"encoding":123},"a_0",[42,7345,7347],{"className":7346,"ariaHidden":129},[128],[42,7348,7350,7354],{"className":7349},[133],[42,7351],{"className":7352,"style":7353},[137],"height:0.5806em;vertical-align:-0.15em;",[42,7355,7357,7360],{"className":7356},[142],[42,7358,2626],{"className":7359},[142,146],[42,7361,7363],{"className":7362},[151],[42,7364,7366,7386],{"className":7365},[155,156],[42,7367,7369,7383],{"className":7368},[160],[42,7370,7372],{"className":7371,"style":937},[164],[42,7373,7374,7377],{"style":6812},[42,7375],{"className":7376,"style":173},[172],[42,7378,7380],{"className":7379},[177,178,179,180],[42,7381,739],{"className":7382},[142,180],[42,7384,199],{"className":7385},[198],[42,7387,7389],{"className":7388},[160],[42,7390,7392],{"className":7391,"style":961},[164],[42,7393],{},[42,7395,7397,7415],{"className":7396,"translate":46},[50],[42,7398,7400],{"className":7399},[54],[56,7401,7402],{"xmlns":58},[61,7403,7404,7412],{},[64,7405,7406],{},[67,7407,7408,7410],{},[70,7409,119],{},[83,7411,739],{},[121,7413,7414],{"encoding":123},"M_0",[42,7416,7418],{"className":7417,"ariaHidden":129},[128],[42,7419,7421,7424],{"className":7420},[133],[42,7422],{"className":7423,"style":918},[137],[42,7425,7427,7430],{"className":7426},[142],[42,7428,119],{"className":7429,"style":147},[142,146],[42,7431,7433],{"className":7432},[151],[42,7434,7436,7456],{"className":7435},[155,156],[42,7437,7439,7453],{"className":7438},[160],[42,7440,7442],{"className":7441,"style":937},[164],[42,7443,7444,7447],{"style":168},[42,7445],{"className":7446,"style":173},[172],[42,7448,7450],{"className":7449},[177,178,179,180],[42,7451,739],{"className":7452},[142,180],[42,7454,199],{"className":7455},[198],[42,7457,7459],{"className":7458},[160],[42,7460,7462],{"className":7461,"style":961},[164],[42,7463],{}," 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= x_0*b_0",[42,7672,7674,7692,7747],{"className":7673,"ariaHidden":129},[128],[42,7675,7677,7680,7683,7686,7689],{"className":7676},[133],[42,7678],{"className":7679,"style":4501},[137],[42,7681,4489],{"className":7682},[142,146],[42,7684],{"className":7685,"style":213},[212],[42,7687,88],{"className":7688},[217],[42,7690],{"className":7691,"style":213},[212],[42,7693,7695,7698,7738,7741,7744],{"className":7694},[133],[42,7696],{"className":7697,"style":6791},[137],[42,7699,7701,7704],{"className":7700},[142],[42,7702,4489],{"className":7703},[142,146],[42,7705,7707],{"className":7706},[151],[42,7708,7710,7730],{"className":7709},[155,156],[42,7711,7713,7727],{"className":7712},[160],[42,7714,7716],{"className":7715,"style":937},[164],[42,7717,7718,7721],{"style":6812},[42,7719],{"className":7720,"style":173},[172],[42,7722,7724],{"className":7723},[177,178,179,180],[42,7725,739],{"className":7726},[142,180],[42,7728,199],{"className":7729},[198],[42,7731,7733],{"className":7732},[160],[42,7734,7736],{"className":7735,"style":961},[164],[42,7737],{},[42,7739],{"className":7740,"style":238},[212],[42,7742,98],{"className":7743},[191],[42,7745],{"className":7746,"style":238},[212],[42,7748,7750,7753],{"className":7749},[133],[42,7751],{"className":7752,"style":6939},[137],[42,7754,7756,7759],{"className":7755},[142],[42,7757,5700],{"className":7758},[142,146],[42,7760,7762],{"className":7761},[151],[42,7763,7765,7785],{"className":7764},[155,156],[42,7766,7768,7782],{"className":7767},[160],[42,7769,7771],{"className":7770,"style":937},[164],[42,7772,7773,7776],{"style":6812},[42,7774],{"className":7775,"style":173},[172],[42,7777,7779],{"className":7778},[177,178,179,180],[42,7780,739],{"className":7781},[142,180],[42,7783,199],{"className":7784},[198],[42,7786,7788],{"className":7787},[160],[42,7789,7791],{"className":7790,"style":961},[164],[42,7792],{},[42,7794,7796,7826],{"className":7795,"translate":46},[50],[42,7797,7799],{"className":7798},[54],[56,7800,7801],{"xmlns":58},[61,7802,7803,7823],{},[64,7804,7805,7807,7809,7815,7817],{},[70,7806,4519],{},[79,7808,88],{},[67,7810,7811,7813],{},[70,7812,4519],{},[83,7814,739],{},[79,7816,98],{},[67,7818,7819,7821],{},[70,7820,5700],{},[83,7822,739],{},[121,7824,7825],{"encoding":123},"y = y_0*b_0",[42,7827,7829,7847,7904],{"className":7828,"ariaHidden":129},[128],[42,7830,7832,7835,7838,7841,7844],{"className":7831},[133],[42,7833],{"className":7834,"style":4531},[137],[42,7836,4519],{"className":7837,"style":4535},[142,146],[42,7839],{"className":7840,"style":213},[212],[42,7842,88],{"className":7843},[217],[42,7845],{"className":7846,"style":213},[212],[42,7848,7850,7854,7895,7898,7901],{"className":7849},[133],[42,7851],{"className":7852,"style":7853},[137],"height:0.6597em;vertical-align:-0.1944em;",[42,7855,7857,7860],{"className":7856},[142],[42,7858,4519],{"className":7859,"style":4535},[142,146],[42,7861,7863],{"className":7862},[151],[42,7864,7866,7887],{"className":7865},[155,156],[42,7867,7869,7884],{"className":7868},[160],[42,7870,7872],{"className":7871,"style":937},[164],[42,7873,7875,7878],{"style":7874},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[42,7876],{"className":7877,"style":173},[172],[42,7879,7881],{"className":7880},[177,178,179,180],[42,7882,739],{"className":7883},[142,180],[42,7885,199],{"className":7886},[198],[42,7888,7890],{"className":7889},[160],[42,7891,7893],{"className":7892,"style":961},[164],[42,7894],{},[42,7896],{"className":7897,"style":238},[212],[42,7899,98],{"className":7900},[191],[42,7902],{"className":7903,"style":238},[212],[42,7905,7907,7910],{"className":7906},[133],[42,7908],{"className":7909,"style":6939},[137],[42,7911,7913,7916],{"className":7912},[142],[42,7914,5700],{"className":7915},[142,146],[42,7917,7919],{"className":7918},[151],[42,7920,7922,7942],{"className":7921},[155,156],[42,7923,7925,7939],{"className":7924},[160],[42,7926,7928],{"className":7927,"style":937},[164],[42,7929,7930,7933],{"style":6812},[42,7931],{"className":7932,"style":173},[172],[42,7934,7936],{"className":7935},[177,178,179,180],[42,7937,739],{"className":7938},[142,180],[42,7940,199],{"className":7941},[198],[42,7943,7945],{"className":7944},[160],[42,7946,7948],{"className":7947,"style":961},[164],[42,7949],{},". The equation becomes ",[42,7952,7954,8000],{"className":7953,"translate":46},[50],[42,7955,7957],{"className":7956},[54],[56,7958,7959],{"xmlns":58},[61,7960,7961,7997],{},[64,7962,7963,7969,7971,7977,7979,7985,7987,7993,7995],{},[67,7964,7965,7967],{},[70,7966,2626],{},[83,7968,739],{},[79,7970,98],{},[67,7972,7973,7975],{},[70,7974,4489],{},[83,7976,739],{},[79,7978,81],{},[67,7980,7981,7983],{},[70,7982,119],{},[83,7984,739],{},[79,7986,98],{},[67,7988,7989,7991],{},[70,7990,4519],{},[83,7992,739],{},[79,7994,88],{},[83,7996,85],{},[121,7998,7999],{"encoding":123},"a_0*x_0 + M_0*y_0 = 1",[42,8001,8003,8058,8114,8169,8224],{"className":8002,"ariaHidden":129},[128],[42,8004,8006,8009,8049,8052,8055],{"className":8005},[133],[42,8007],{"className":8008,"style":6791},[137],[42,8010,8012,8015],{"className":8011},[142],[42,8013,2626],{"className":8014},[142,146],[42,8016,8018],{"className":8017},[151],[42,8019,8021,8041],{"className":8020},[155,156],[42,8022,8024,8038],{"className":8023},[160],[42,8025,8027],{"className":8026,"style":937},[164],[42,8028,8029,8032],{"style":6812},[42,8030],{"className":8031,"style":173},[172],[42,8033,8035],{"className":8034},[177,178,179,180],[42,8036,739],{"className":8037},[142,180],[42,8039,199],{"className":8040},[198],[42,8042,8044],{"className":8043},[160],[42,8045,8047],{"className":8046,"style":961},[164],[42,8048],{},[42,8050],{"className":8051,"style":238},[212],[42,8053,98],{"className":8054},[191],[42,8056],{"className":8057,"style":238},[212],[42,8059,8061,8065,8105,8108,8111],{"className":8060},[133],[42,8062],{"className":8063,"style":8064},[137],"height:0.7333em;vertical-align:-0.15em;",[42,8066,8068,8071],{"className":8067},[142],[42,8069,4489],{"className":8070},[142,146],[42,8072,8074],{"className":8073},[151],[42,8075,8077,8097],{"className":8076},[155,156],[42,8078,8080,8094],{"className":8079},[160],[42,8081,8083],{"className":8082,"style":937},[164],[42,8084,8085,8088],{"style":6812},[42,8086],{"className":8087,"style":173},[172],[42,8089,8091],{"className":8090},[177,178,179,180],[42,8092,739],{"className":8093},[142,180],[42,8095,199],{"className":8096},[198],[42,8098,8100],{"className":8099},[160],[42,8101,8103],{"className":8102,"style":961},[164],[42,8104],{},[42,8106],{"className":8107,"style":238},[212],[42,8109,81],{"className":8110},[191],[42,8112],{"className":8113,"style":238},[212],[42,8115,8117,8120,8160,8163,8166],{"className":8116},[133],[42,8118],{"className":8119,"style":918},[137],[42,8121,8123,8126],{"className":8122},[142],[42,8124,119],{"className":8125,"style":147},[142,146],[42,8127,8129],{"className":8128},[151],[42,8130,8132,8152],{"className":8131},[155,156],[42,8133,8135,8149],{"className":8134},[160],[42,8136,8138],{"className":8137,"style":937},[164],[42,8139,8140,8143],{"style":168},[42,8141],{"className":8142,"style":173},[172],[42,8144,8146],{"className":8145},[177,178,179,180],[42,8147,739],{"className":8148},[142,180],[42,8150,199],{"className":8151},[198],[42,8153,8155],{"className":8154},[160],[42,8156,8158],{"className":8157,"style":961},[164],[42,8159],{},[42,8161],{"className":8162,"style":238},[212],[42,8164,98],{"className":8165},[191],[42,8167],{"className":8168,"style":238},[212],[42,8170,8172,8175,8215,8218,8221],{"className":8171},[133],[42,8173],{"className":8174,"style":4531},[137],[42,8176,8178,8181],{"className":8177},[142],[42,8179,4519],{"className":8180,"style":4535},[142,146],[42,8182,8184],{"className":8183},[151],[42,8185,8187,8207],{"className":8186},[155,156],[42,8188,8190,8204],{"className":8189},[160],[42,8191,8193],{"className":8192,"style":937},[164],[42,8194,8195,8198],{"style":7874},[42,8196],{"className":8197,"style":173},[172],[42,8199,8201],{"className":8200},[177,178,179,180],[42,8202,739],{"className":8203},[142,180],[42,8205,199],{"className":8206},[198],[42,8208,8210],{"className":8209},[160],[42,8211,8213],{"className":8212,"style":961},[164],[42,8214],{},[42,8216],{"className":8217,"style":213},[212],[42,8219,88],{"className":8220},[217],[42,8222],{"className":8223,"style":213},[212],[42,8225,8227,8230],{"className":8226},[133],[42,8228],{"className":8229,"style":708},[137],[42,8231,85],{"className":8232},[142],", or equivalently ",[42,8235,8237,8291],{"className":8236,"translate":46},[50],[42,8238,8240],{"className":8239},[54],[56,8241,8242],{"xmlns":58},[61,8243,8244,8288],{},[64,8245,8246,8252,8254,8260,8262,8264,8266,8268,8270,8278,8280,8286],{},[67,8247,8248,8250],{},[70,8249,2626],{},[83,8251,739],{},[79,8253,98],{},[67,8255,8256,8258],{},[70,8257,4489],{},[83,8259,739],{},[79,8261,4717],{},[83,8263,85],{},[212,8265],{},[212,8267],{"width":5973},[79,8269,92],{"stretchy":91},[64,8271,8272,8274,8276],{},[70,8273,1250],{"mathvariant":115},[70,8275,1253],{"mathvariant":115},[70,8277,1256],{"mathvariant":115},[212,8279],{"width":4749},[67,8281,8282,8284],{},[70,8283,119],{},[83,8285,739],{},[79,8287,112],{"stretchy":91},[121,8289,8290],{"encoding":123},"a_0*x_0 ≡ 1 \\pmod{M_0}",[42,8292,8294,8349,8405,8420],{"className":8293,"ariaHidden":129},[128],[42,8295,8297,8300,8340,8343,8346],{"className":8296},[133],[42,8298],{"className":8299,"style":6791},[137],[42,8301,8303,8306],{"className":8302},[142],[42,8304,2626],{"className":8305},[142,146],[42,8307,8309],{"className":8308},[151],[42,8310,8312,8332],{"className":8311},[155,156],[42,8313,8315,8329],{"className":8314},[160],[42,8316,8318],{"className":8317,"style":937},[164],[42,8319,8320,8323],{"style":6812},[42,8321],{"className":8322,"style":173},[172],[42,8324,8326],{"className":8325},[177,178,179,180],[42,8327,739],{"className":8328},[142,180],[42,8330,199],{"className":8331},[198],[42,8333,8335],{"className":8334},[160],[42,8336,8338],{"className":8337,"style":961},[164],[42,8339],{},[42,8341],{"className":8342,"style":238},[212],[42,8344,98],{"className":8345},[191],[42,8347],{"className":8348,"style":238},[212],[42,8350,8352,8356,8396,8399,8402],{"className":8351},[133],[42,8353],{"className":8354,"style":8355},[137],"height:0.6138em;vertical-align:-0.15em;",[42,8357,8359,8362],{"className":8358},[142],[42,8360,4489],{"className":8361},[142,146],[42,8363,8365],{"className":8364},[151],[42,8366,8368,8388],{"className":8367},[155,156],[42,8369,8371,8385],{"className":8370},[160],[42,8372,8374],{"className":8373,"style":937},[164],[42,8375,8376,8379],{"style":6812},[42,8377],{"className":8378,"style":173},[172],[42,8380,8382],{"className":8381},[177,178,179,180],[42,8383,739],{"className":8384},[142,180],[42,8386,199],{"className":8387},[198],[42,8389,8391],{"className":8390},[160],[42,8392,8394],{"className":8393,"style":961},[164],[42,8395],{},[42,8397],{"className":8398,"style":213},[212],[42,8400,4717],{"className":8401},[217],[42,8403],{"className":8404,"style":213},[212],[42,8406,8408,8411,8414,8417],{"className":8407},[133],[42,8409],{"className":8410,"style":708},[137],[42,8412,85],{"className":8413},[142],[42,8415],{"className":8416},[212,1429],[42,8418],{"className":8419,"style":6047},[212],[42,8421,8423,8426,8429,8438,8441,8481],{"className":8422},[133],[42,8424],{"className":8425,"style":227},[137],[42,8427,92],{"className":8428},[231],[42,8430,8432],{"className":8431},[142],[42,8433,8435],{"className":8434},[142],[42,8436,1451],{"className":8437},[142,1450],[42,8439],{"className":8440,"style":4899},[212],[42,8442,8444,8447],{"className":8443},[142],[42,8445,119],{"className":8446,"style":147},[142,146],[42,8448,8450],{"className":8449},[151],[42,8451,8453,8473],{"className":8452},[155,156],[42,8454,8456,8470],{"className":8455},[160],[42,8457,8459],{"className":8458,"style":937},[164],[42,8460,8461,8464],{"style":168},[42,8462],{"className":8463,"style":173},[172],[42,8465,8467],{"className":8466},[177,178,179,180],[42,8468,739],{"className":8469},[142,180],[42,8471,199],{"className":8472},[198],[42,8474,8476],{"className":8475},[160],[42,8477,8479],{"className":8478,"style":961},[164],[42,8480],{},[42,8482,112],{"className":8483},[316],[15,8485,8486,8487,413],{},"Finally, we use modular inversion to compute ",[42,8488,8490,8508],{"className":8489,"translate":46},[50],[42,8491,8493],{"className":8492},[54],[56,8494,8495],{"xmlns":58},[61,8496,8497,8505],{},[64,8498,8499],{},[67,8500,8501,8503],{},[70,8502,4489],{},[83,8504,739],{},[121,8506,8507],{"encoding":123},"x_0",[42,8509,8511],{"className":8510,"ariaHidden":129},[128],[42,8512,8514,8517],{"className":8513},[133],[42,8515],{"className":8516,"style":7353},[137],[42,8518,8520,8523],{"className":8519},[142],[42,8521,4489],{"className":8522},[142,146],[42,8524,8526],{"className":8525},[151],[42,8527,8529,8549],{"className":8528},[155,156],[42,8530,8532,8546],{"className":8531},[160],[42,8533,8535],{"className":8534,"style":937},[164],[42,8536,8537,8540],{"style":6812},[42,8538],{"className":8539,"style":173},[172],[42,8541,8543],{"className":8542},[177,178,179,180],[42,8544,739],{"className":8545},[142,180],[42,8547,199],{"className":8548},[198],[42,8550,8552],{"className":8551},[160],[42,8553,8555],{"className":8554,"style":961},[164],[42,8556],{},[19,8558,8559],{},[15,8560,8561,8564,8565,596,8593,8621,8622,8674,8675,8703,8704,413],{},[24,8562,8563],{},"Theorem:"," If ",[42,8566,8568,8581],{"className":8567,"translate":46},[50],[42,8569,8571],{"className":8570},[54],[56,8572,8573],{"xmlns":58},[61,8574,8575,8579],{},[64,8576,8577],{},[70,8578,2626],{},[121,8580,2626],{"encoding":123},[42,8582,8584],{"className":8583,"ariaHidden":129},[128],[42,8585,8587,8590],{"className":8586},[133],[42,8588],{"className":8589,"style":4501},[137],[42,8591,2626],{"className":8592},[142,146],[42,8594,8596,8609],{"className":8595,"translate":46},[50],[42,8597,8599],{"className":8598},[54],[56,8600,8601],{"xmlns":58},[61,8602,8603,8607],{},[64,8604,8605],{},[70,8606,119],{},[121,8608,119],{"encoding":123},[42,8610,8612],{"className":8611,"ariaHidden":129},[128],[42,8613,8615,8618],{"className":8614},[133],[42,8616],{"className":8617,"style":351},[137],[42,8619,119],{"className":8620,"style":147},[142,146]," are coprime integers and ",[42,8623,8625,8643],{"className":8624,"translate":46},[50],[42,8626,8628],{"className":8627},[54],[56,8629,8630],{"xmlns":58},[61,8631,8632,8640],{},[64,8633,8634,8636,8638],{},[70,8635,119],{},[79,8637,2773],{},[83,8639,85],{},[121,8641,8642],{"encoding":123},"M > 1",[42,8644,8646,8665],{"className":8645,"ariaHidden":129},[128],[42,8647,8649,8653,8656,8659,8662],{"className":8648},[133],[42,8650],{"className":8651,"style":8652},[137],"height:0.7224em;vertical-align:-0.0391em;",[42,8654,119],{"className":8655,"style":147},[142,146],[42,8657],{"className":8658,"style":213},[212],[42,8660,2773],{"className":8661},[217],[42,8663],{"className":8664,"style":213},[212],[42,8666,8668,8671],{"className":8667},[133],[42,8669],{"className":8670,"style":708},[137],[42,8672,85],{"className":8673},[142],", there exists a unique inverse of ",[42,8676,8678,8691],{"className":8677,"translate":46},[50],[42,8679,8681],{"className":8680},[54],[56,8682,8683],{"xmlns":58},[61,8684,8685,8689],{},[64,8686,8687],{},[70,8688,2626],{},[121,8690,2626],{"encoding":123},[42,8692,8694],{"className":8693,"ariaHidden":129},[128],[42,8695,8697,8700],{"className":8696},[133],[42,8698],{"className":8699,"style":4501},[137],[42,8701,2626],{"className":8702},[142,146]," modulo ",[42,8705,8707,8720],{"className":8706,"translate":46},[50],[42,8708,8710],{"className":8709},[54],[56,8711,8712],{"xmlns":58},[61,8713,8714,8718],{},[64,8715,8716],{},[70,8717,119],{},[121,8719,119],{"encoding":123},[42,8721,8723],{"className":8722,"ariaHidden":129},[128],[42,8724,8726,8729],{"className":8725},[133],[42,8727],{"className":8728,"style":351},[137],[42,8730,119],{"className":8731,"style":147},[142,146],[15,8733,8734],{},"This theorem can be proven as follows:",[15,8736,8737,8738,596,8766,8794,8795,596,8824,8854,8855,8972,8973,596,9001,9029,9030,778,9080,9219,9220,9248,9249,9375,9376,9404,9405,8703,9433,413],{},"Since ",[42,8739,8741,8754],{"className":8740,"translate":46},[50],[42,8742,8744],{"className":8743},[54],[56,8745,8746],{"xmlns":58},[61,8747,8748,8752],{},[64,8749,8750],{},[70,8751,2626],{},[121,8753,2626],{"encoding":123},[42,8755,8757],{"className":8756,"ariaHidden":129},[128],[42,8758,8760,8763],{"className":8759},[133],[42,8761],{"className":8762,"style":4501},[137],[42,8764,2626],{"className":8765},[142,146],[42,8767,8769,8782],{"className":8768,"translate":46},[50],[42,8770,8772],{"className":8771},[54],[56,8773,8774],{"xmlns":58},[61,8775,8776,8780],{},[64,8777,8778],{},[70,8779,119],{},[121,8781,119],{"encoding":123},[42,8783,8785],{"className":8784,"ariaHidden":129},[128],[42,8786,8788,8791],{"className":8787},[133],[42,8789],{"className":8790,"style":351},[137],[42,8792,119],{"className":8793,"style":147},[142,146]," are coprime, there exist integers ",[42,8796,8798,8812],{"className":8797,"translate":46},[50],[42,8799,8801],{"className":8800},[54],[56,8802,8803],{"xmlns":58},[61,8804,8805,8810],{},[64,8806,8807],{},[70,8808,8809],{},"s",[121,8811,8809],{"encoding":123},[42,8813,8815],{"className":8814,"ariaHidden":129},[128],[42,8816,8818,8821],{"className":8817},[133],[42,8819],{"className":8820,"style":4501},[137],[42,8822,8809],{"className":8823},[142,146],[42,8825,8827,8841],{"className":8826,"translate":46},[50],[42,8828,8830],{"className":8829},[54],[56,8831,8832],{"xmlns":58},[61,8833,8834,8839],{},[64,8835,8836],{},[70,8837,8838],{},"t",[121,8840,8838],{"encoding":123},[42,8842,8844],{"className":8843,"ariaHidden":129},[128],[42,8845,8847,8851],{"className":8846},[133],[42,8848],{"className":8849,"style":8850},[137],"height:0.6151em;",[42,8852,8838],{"className":8853},[142,146]," such that ",[42,8856,8858,8888],{"className":8857,"translate":46},[50],[42,8859,8861],{"className":8860},[54],[56,8862,8863],{"xmlns":58},[61,8864,8865,8885],{},[64,8866,8867,8869,8871,8873,8875,8877,8879,8881,8883],{},[70,8868,8809],{},[79,8870,98],{},[70,8872,2626],{},[79,8874,81],{},[70,8876,8838],{},[79,8878,98],{},[70,8880,119],{},[79,8882,88],{},[83,8884,85],{},[121,8886,8887],{"encoding":123},"s*a + t*M = 1",[42,8889,8891,8909,8927,8945,8963],{"className":8890,"ariaHidden":129},[128],[42,8892,8894,8897,8900,8903,8906],{"className":8893},[133],[42,8895],{"className":8896,"style":2661},[137],[42,8898,8809],{"className":8899},[142,146],[42,8901],{"className":8902,"style":238},[212],[42,8904,98],{"className":8905},[191],[42,8907],{"className":8908,"style":238},[212],[42,8910,8912,8915,8918,8921,8924],{"className":8911},[133],[42,8913],{"className":8914,"style":5033},[137],[42,8916,2626],{"className":8917},[142,146],[42,8919],{"className":8920,"style":238},[212],[42,8922,81],{"className":8923},[191],[42,8925],{"className":8926,"style":238},[212],[42,8928,8930,8933,8936,8939,8942],{"className":8929},[133],[42,8931],{"className":8932,"style":8850},[137],[42,8934,8838],{"className":8935},[142,146],[42,8937],{"className":8938,"style":238},[212],[42,8940,98],{"className":8941},[191],[42,8943],{"className":8944,"style":238},[212],[42,8946,8948,8951,8954,8957,8960],{"className":8947},[133],[42,8949],{"className":8950,"style":351},[137],[42,8952,119],{"className":8953,"style":147},[142,146],[42,8955],{"className":8956,"style":213},[212],[42,8958,88],{"className":8959},[217],[42,8961],{"className":8962,"style":213},[212],[42,8964,8966,8969],{"className":8965},[133],[42,8967],{"className":8968,"style":708},[137],[42,8970,85],{"className":8971},[142],". (If ",[42,8974,8976,8989],{"className":8975,"translate":46},[50],[42,8977,8979],{"className":8978},[54],[56,8980,8981],{"xmlns":58},[61,8982,8983,8987],{},[64,8984,8985],{},[70,8986,2626],{},[121,8988,2626],{"encoding":123},[42,8990,8992],{"className":8991,"ariaHidden":129},[128],[42,8993,8995,8998],{"className":8994},[133],[42,8996],{"className":8997,"style":4501},[137],[42,8999,2626],{"className":9000},[142,146],[42,9002,9004,9017],{"className":9003,"translate":46},[50],[42,9005,9007],{"className":9006},[54],[56,9008,9009],{"xmlns":58},[61,9010,9011,9015],{},[64,9012,9013],{},[70,9014,119],{},[121,9016,119],{"encoding":123},[42,9018,9020],{"className":9019,"ariaHidden":129},[128],[42,9021,9023,9026],{"className":9022},[133],[42,9024],{"className":9025,"style":351},[137],[42,9027,119],{"className":9028,"style":147},[142,146]," have a common divisor ",[42,9031,9033,9050],{"className":9032,"translate":46},[50],[42,9034,9036],{"className":9035},[54],[56,9037,9038],{"xmlns":58},[61,9039,9040,9048],{},[64,9041,9042,9044,9046],{},[70,9043,1256],{},[79,9045,2773],{},[83,9047,85],{},[121,9049,2778],{"encoding":123},[42,9051,9053,9071],{"className":9052,"ariaHidden":129},[128],[42,9054,9056,9059,9062,9065,9068],{"className":9055},[133],[42,9057],{"className":9058,"style":2788},[137],[42,9060,1256],{"className":9061},[142,146],[42,9063],{"className":9064,"style":213},[212],[42,9066,2773],{"className":9067},[217],[42,9069],{"className":9070,"style":213},[212],[42,9072,9074,9077],{"className":9073},[133],[42,9075],{"className":9076,"style":708},[137],[42,9078,85],{"className":9079},[142],[42,9081,9083,9117],{"className":9082,"translate":46},[50],[42,9084,9086],{"className":9085},[54],[56,9087,9088],{"xmlns":58},[61,9089,9090,9114],{},[64,9091,9092,9094,9096,9098,9100,9102,9104,9106,9108,9110,9112],{},[70,9093,8809],{},[79,9095,98],{},[70,9097,2626],{},[79,9099,81],{},[70,9101,8838],{},[79,9103,98],{},[70,9105,119],{},[79,9107,88],{},[70,9109,1256],{},[79,9111,98],{},[70,9113,1513],{},[121,9115,9116],{"encoding":123},"s*a + t*M = d*k",[42,9118,9120,9138,9156,9174,9192,9210],{"className":9119,"ariaHidden":129},[128],[42,9121,9123,9126,9129,9132,9135],{"className":9122},[133],[42,9124],{"className":9125,"style":2661},[137],[42,9127,8809],{"className":9128},[142,146],[42,9130],{"className":9131,"style":238},[212],[42,9133,98],{"className":9134},[191],[42,9136],{"className":9137,"style":238},[212],[42,9139,9141,9144,9147,9150,9153],{"className":9140},[133],[42,9142],{"className":9143,"style":5033},[137],[42,9145,2626],{"className":9146},[142,146],[42,9148],{"className":9149,"style":238},[212],[42,9151,81],{"className":9152},[191],[42,9154],{"className":9155,"style":238},[212],[42,9157,9159,9162,9165,9168,9171],{"className":9158},[133],[42,9160],{"className":9161,"style":8850},[137],[42,9163,8838],{"className":9164},[142,146],[42,9166],{"className":9167,"style":238},[212],[42,9169,98],{"className":9170},[191],[42,9172],{"className":9173,"style":238},[212],[42,9175,9177,9180,9183,9186,9189],{"className":9176},[133],[42,9178],{"className":9179,"style":351},[137],[42,9181,119],{"className":9182,"style":147},[142,146],[42,9184],{"className":9185,"style":213},[212],[42,9187,88],{"className":9188},[217],[42,9190],{"className":9191,"style":213},[212],[42,9193,9195,9198,9201,9204,9207],{"className":9194},[133],[42,9196],{"className":9197,"style":1440},[137],[42,9199,1256],{"className":9200},[142,146],[42,9202],{"className":9203,"style":238},[212],[42,9205,98],{"className":9206},[191],[42,9208],{"className":9209,"style":238},[212],[42,9211,9213,9216],{"className":9212},[133],[42,9214],{"className":9215,"style":1440},[137],[42,9217,1513],{"className":9218,"style":1572},[142,146],", which cannot equal 1). Taking the equation modulo ",[42,9221,9223,9236],{"className":9222,"translate":46},[50],[42,9224,9226],{"className":9225},[54],[56,9227,9228],{"xmlns":58},[61,9229,9230,9234],{},[64,9231,9232],{},[70,9233,119],{},[121,9235,119],{"encoding":123},[42,9237,9239],{"className":9238,"ariaHidden":129},[128],[42,9240,9242,9245],{"className":9241},[133],[42,9243],{"className":9244,"style":351},[137],[42,9246,119],{"className":9247,"style":147},[142,146],", we get ",[42,9250,9252,9294],{"className":9251,"translate":46},[50],[42,9253,9255],{"className":9254},[54],[56,9256,9257],{"xmlns":58},[61,9258,9259,9291],{},[64,9260,9261,9263,9265,9267,9269,9271,9273,9275,9277,9285,9287,9289],{},[70,9262,8809],{},[79,9264,98],{},[70,9266,2626],{},[79,9268,4717],{},[83,9270,85],{},[212,9272],{},[212,9274],{"width":5973},[79,9276,92],{"stretchy":91},[64,9278,9279,9281,9283],{},[70,9280,1250],{"mathvariant":115},[70,9282,1253],{"mathvariant":115},[70,9284,1256],{"mathvariant":115},[212,9286],{"width":4749},[70,9288,119],{},[79,9290,112],{"stretchy":91},[121,9292,9293],{"encoding":123},"s*a ≡ 1 \\pmod{M}",[42,9295,9297,9315,9333,9348],{"className":9296,"ariaHidden":129},[128],[42,9298,9300,9303,9306,9309,9312],{"className":9299},[133],[42,9301],{"className":9302,"style":2661},[137],[42,9304,8809],{"className":9305},[142,146],[42,9307],{"className":9308,"style":238},[212],[42,9310,98],{"className":9311},[191],[42,9313],{"className":9314,"style":238},[212],[42,9316,9318,9321,9324,9327,9330],{"className":9317},[133],[42,9319],{"className":9320,"style":4934},[137],[42,9322,2626],{"className":9323},[142,146],[42,9325],{"className":9326,"style":213},[212],[42,9328,4717],{"className":9329},[217],[42,9331],{"className":9332,"style":213},[212],[42,9334,9336,9339,9342,9345],{"className":9335},[133],[42,9337],{"className":9338,"style":708},[137],[42,9340,85],{"className":9341},[142],[42,9343],{"className":9344},[212,1429],[42,9346],{"className":9347,"style":6047},[212],[42,9349,9351,9354,9357,9366,9369,9372],{"className":9350},[133],[42,9352],{"className":9353,"style":227},[137],[42,9355,92],{"className":9356},[231],[42,9358,9360],{"className":9359},[142],[42,9361,9363],{"className":9362},[142],[42,9364,1451],{"className":9365},[142,1450],[42,9367],{"className":9368,"style":4899},[212],[42,9370,119],{"className":9371,"style":147},[142,146],[42,9373,112],{"className":9374},[316],". Thus, ",[42,9377,9379,9392],{"className":9378,"translate":46},[50],[42,9380,9382],{"className":9381},[54],[56,9383,9384],{"xmlns":58},[61,9385,9386,9390],{},[64,9387,9388],{},[70,9389,8809],{},[121,9391,8809],{"encoding":123},[42,9393,9395],{"className":9394,"ariaHidden":129},[128],[42,9396,9398,9401],{"className":9397},[133],[42,9399],{"className":9400,"style":4501},[137],[42,9402,8809],{"className":9403},[142,146]," is the modular inverse of ",[42,9406,9408,9421],{"className":9407,"translate":46},[50],[42,9409,9411],{"className":9410},[54],[56,9412,9413],{"xmlns":58},[61,9414,9415,9419],{},[64,9416,9417],{},[70,9418,2626],{},[121,9420,2626],{"encoding":123},[42,9422,9424],{"className":9423,"ariaHidden":129},[128],[42,9425,9427,9430],{"className":9426},[133],[42,9428],{"className":9429,"style":4501},[137],[42,9431,2626],{"className":9432},[142,146],[42,9434,9436,9449],{"className":9435,"translate":46},[50],[42,9437,9439],{"className":9438},[54],[56,9440,9441],{"xmlns":58},[61,9442,9443,9447],{},[64,9444,9445],{},[70,9446,119],{},[121,9448,119],{"encoding":123},[42,9450,9452],{"className":9451,"ariaHidden":129},[128],[42,9453,9455,9458],{"className":9454},[133],[42,9456],{"className":9457,"style":351},[137],[42,9459,119],{"className":9460,"style":147},[142,146],[15,9462,5604,9463,9532,9533,8703,9561,9589,9590,9808,9809,413],{},[42,9464,9466,9486],{"className":9465,"translate":46},[50],[42,9467,9469],{"className":9468},[54],[56,9470,9471],{"xmlns":58},[61,9472,9473,9483],{},[64,9474,9475],{},[9476,9477,9478,9480],"mover",{"accent":129},[70,9479,2626],{},[79,9481,9482],{},"ˉ",[121,9484,9485],{"encoding":123},"\\bar{a}",[42,9487,9489],{"className":9488,"ariaHidden":129},[128],[42,9490,9492,9496],{"className":9491},[133],[42,9493],{"className":9494,"style":9495},[137],"height:0.5678em;",[42,9497,9500],{"className":9498},[142,9499],"accent",[42,9501,9503],{"className":9502},[155],[42,9504,9506],{"className":9505},[160],[42,9507,9509,9519],{"className":9508,"style":9495},[164],[42,9510,9512,9516],{"style":9511},"top:-3em;",[42,9513],{"className":9514,"style":9515},[172],"height:3em;",[42,9517,2626],{"className":9518},[142,146],[42,9520,9521,9524],{"style":9511},[42,9522],{"className":9523,"style":9515},[172],[42,9525,9529],{"className":9526,"style":9528},[9527],"accent-body","left:-0.25em;",[42,9530,9482],{"className":9531},[142]," denote the modular inverse of ",[42,9534,9536,9549],{"className":9535,"translate":46},[50],[42,9537,9539],{"className":9538},[54],[56,9540,9541],{"xmlns":58},[61,9542,9543,9547],{},[64,9544,9545],{},[70,9546,2626],{},[121,9548,2626],{"encoding":123},[42,9550,9552],{"className":9551,"ariaHidden":129},[128],[42,9553,9555,9558],{"className":9554},[133],[42,9556],{"className":9557,"style":4501},[137],[42,9559,2626],{"className":9560},[142,146],[42,9562,9564,9577],{"className":9563,"translate":46},[50],[42,9565,9567],{"className":9566},[54],[56,9568,9569],{"xmlns":58},[61,9570,9571,9575],{},[64,9572,9573],{},[70,9574,119],{},[121,9576,119],{"encoding":123},[42,9578,9580],{"className":9579,"ariaHidden":129},[128],[42,9581,9583,9586],{"className":9582},[133],[42,9584],{"className":9585,"style":351},[137],[42,9587,119],{"className":9588,"style":147},[142,146],", so ",[42,9591,9593,9643],{"className":9592,"translate":46},[50],[42,9594,9596],{"className":9595},[54],[56,9597,9598],{"xmlns":58},[61,9599,9600,9640],{},[64,9601,9602,9608,9610,9616,9618,9620,9622,9630,9632,9638],{},[67,9603,9604,9606],{},[70,9605,4489],{},[83,9607,739],{},[79,9609,4717],{},[9476,9611,9612,9614],{"accent":129},[70,9613,2626],{},[79,9615,9482],{},[212,9617],{},[212,9619],{"width":5973},[79,9621,92],{"stretchy":91},[64,9623,9624,9626,9628],{},[70,9625,1250],{"mathvariant":115},[70,9627,1253],{"mathvariant":115},[70,9629,1256],{"mathvariant":115},[212,9631],{"width":4749},[67,9633,9634,9636],{},[70,9635,119],{},[83,9637,739],{},[79,9639,112],{"stretchy":91},[121,9641,9642],{"encoding":123},"x_0 ≡ \\bar{a} \\pmod{M_0}",[42,9644,9646,9701,9744],{"className":9645,"ariaHidden":129},[128],[42,9647,9649,9652,9692,9695,9698],{"className":9648},[133],[42,9650],{"className":9651,"style":8355},[137],[42,9653,9655,9658],{"className":9654},[142],[42,9656,4489],{"className":9657},[142,146],[42,9659,9661],{"className":9660},[151],[42,9662,9664,9684],{"className":9663},[155,156],[42,9665,9667,9681],{"className":9666},[160],[42,9668,9670],{"className":9669,"style":937},[164],[42,9671,9672,9675],{"style":6812},[42,9673],{"className":9674,"style":173},[172],[42,9676,9678],{"className":9677},[177,178,179,180],[42,9679,739],{"className":9680},[142,180],[42,9682,199],{"className":9683},[198],[42,9685,9687],{"className":9686},[160],[42,9688,9690],{"className":9689,"style":961},[164],[42,9691],{},[42,9693],{"className":9694,"style":213},[212],[42,9696,4717],{"className":9697},[217],[42,9699],{"className":9700,"style":213},[212],[42,9702,9704,9707,9738,9741],{"className":9703},[133],[42,9705],{"className":9706,"style":9495},[137],[42,9708,9710],{"className":9709},[142,9499],[42,9711,9713],{"className":9712},[155],[42,9714,9716],{"className":9715},[160],[42,9717,9719,9727],{"className":9718,"style":9495},[164],[42,9720,9721,9724],{"style":9511},[42,9722],{"className":9723,"style":9515},[172],[42,9725,2626],{"className":9726},[142,146],[42,9728,9729,9732],{"style":9511},[42,9730],{"className":9731,"style":9515},[172],[42,9733,9735],{"className":9734,"style":9528},[9527],[42,9736,9482],{"className":9737},[142],[42,9739],{"className":9740},[212,1429],[42,9742],{"className":9743,"style":6047},[212],[42,9745,9747,9750,9753,9762,9765,9805],{"className":9746},[133],[42,9748],{"className":9749,"style":227},[137],[42,9751,92],{"className":9752},[231],[42,9754,9756],{"className":9755},[142],[42,9757,9759],{"className":9758},[142],[42,9760,1451],{"className":9761},[142,1450],[42,9763],{"className":9764,"style":4899},[212],[42,9766,9768,9771],{"className":9767},[142],[42,9769,119],{"className":9770,"style":147},[142,146],[42,9772,9774],{"className":9773},[151],[42,9775,9777,9797],{"className":9776},[155,156],[42,9778,9780,9794],{"className":9779},[160],[42,9781,9783],{"className":9782,"style":937},[164],[42,9784,9785,9788],{"style":168},[42,9786],{"className":9787,"style":173},[172],[42,9789,9791],{"className":9790},[177,178,179,180],[42,9792,739],{"className":9793},[142,180],[42,9795,199],{"className":9796},[198],[42,9798,9800],{"className":9799},[160],[42,9801,9803],{"className":9802,"style":961},[164],[42,9804],{},[42,9806,112],{"className":9807},[316],", meaning ",[42,9810,9812,9850],{"className":9811,"translate":46},[50],[42,9813,9815],{"className":9814},[54],[56,9816,9817],{"xmlns":58},[61,9818,9819,9847],{},[64,9820,9821,9827,9829,9835,9837,9839,9841],{},[67,9822,9823,9825],{},[70,9824,4489],{},[83,9826,739],{},[79,9828,4717],{},[9476,9830,9831,9833],{"accent":129},[70,9832,2626],{},[79,9834,9482],{},[79,9836,81],{},[70,9838,1513],{},[79,9840,98],{},[67,9842,9843,9845],{},[70,9844,119],{},[83,9846,739],{},[121,9848,9849],{"encoding":123},"x_0 ≡ \\bar{a} + k*M_0",[42,9851,9853,9908,9954,9972],{"className":9852,"ariaHidden":129},[128],[42,9854,9856,9859,9899,9902,9905],{"className":9855},[133],[42,9857],{"className":9858,"style":8355},[137],[42,9860,9862,9865],{"className":9861},[142],[42,9863,4489],{"className":9864},[142,146],[42,9866,9868],{"className":9867},[151],[42,9869,9871,9891],{"className":9870},[155,156],[42,9872,9874,9888],{"className":9873},[160],[42,9875,9877],{"className":9876,"style":937},[164],[42,9878,9879,9882],{"style":6812},[42,9880],{"className":9881,"style":173},[172],[42,9883,9885],{"className":9884},[177,178,179,180],[42,9886,739],{"className":9887},[142,180],[42,9889,199],{"className":9890},[198],[42,9892,9894],{"className":9893},[160],[42,9895,9897],{"className":9896,"style":961},[164],[42,9898],{},[42,9900],{"className":9901,"style":213},[212],[42,9903,4717],{"className":9904},[217],[42,9906],{"className":9907,"style":213},[212],[42,9909,9911,9914,9945,9948,9951],{"className":9910},[133],[42,9912],{"className":9913,"style":5033},[137],[42,9915,9917],{"className":9916},[142,9499],[42,9918,9920],{"className":9919},[155],[42,9921,9923],{"className":9922},[160],[42,9924,9926,9934],{"className":9925,"style":9495},[164],[42,9927,9928,9931],{"style":9511},[42,9929],{"className":9930,"style":9515},[172],[42,9932,2626],{"className":9933},[142,146],[42,9935,9936,9939],{"style":9511},[42,9937],{"className":9938,"style":9515},[172],[42,9940,9942],{"className":9941,"style":9528},[9527],[42,9943,9482],{"className":9944},[142],[42,9946],{"className":9947,"style":238},[212],[42,9949,81],{"className":9950},[191],[42,9952],{"className":9953,"style":238},[212],[42,9955,9957,9960,9963,9966,9969],{"className":9956},[133],[42,9958],{"className":9959,"style":1440},[137],[42,9961,1513],{"className":9962,"style":1572},[142,146],[42,9964],{"className":9965,"style":238},[212],[42,9967,98],{"className":9968},[191],[42,9970],{"className":9971,"style":238},[212],[42,9973,9975,9978],{"className":9974},[133],[42,9976],{"className":9977,"style":918},[137],[42,9979,9981,9984],{"className":9980},[142],[42,9982,119],{"className":9983,"style":147},[142,146],[42,9985,9987],{"className":9986},[151],[42,9988,9990,10010],{"className":9989},[155,156],[42,9991,9993,10007],{"className":9992},[160],[42,9994,9996],{"className":9995,"style":937},[164],[42,9997,9998,10001],{"style":168},[42,9999],{"className":10000,"style":173},[172],[42,10002,10004],{"className":10003},[177,178,179,180],[42,10005,739],{"className":10006},[142,180],[42,10008,199],{"className":10009},[198],[42,10011,10013],{"className":10012},[160],[42,10014,10016],{"className":10015,"style":961},[164],[42,10017],{},[15,10019,10020],{},"For example:",[15,10022,10023,10024,10152],{},"To solve ",[42,10025,10027,10071],{"className":10026,"translate":46},[50],[42,10028,10030],{"className":10029},[54],[56,10031,10032],{"xmlns":58},[61,10033,10034,10068],{},[64,10035,10036,10038,10040,10042,10044,10047,10049,10051,10053,10061,10063,10066],{},[83,10037,3839],{},[79,10039,98],{},[70,10041,4489],{},[79,10043,4717],{},[83,10045,10046],{},"7",[212,10048],{},[212,10050],{"width":5973},[79,10052,92],{"stretchy":91},[64,10054,10055,10057,10059],{},[70,10056,1250],{"mathvariant":115},[70,10058,1253],{"mathvariant":115},[70,10060,1256],{"mathvariant":115},[212,10062],{"width":4749},[83,10064,10065],{},"9",[79,10067,112],{"stretchy":91},[121,10069,10070],{"encoding":123},"5*x ≡ 7 \\pmod{9}",[42,10072,10074,10092,10110,10125],{"className":10073,"ariaHidden":129},[128],[42,10075,10077,10080,10083,10086,10089],{"className":10076},[133],[42,10078],{"className":10079,"style":708},[137],[42,10081,3839],{"className":10082},[142],[42,10084],{"className":10085,"style":238},[212],[42,10087,98],{"className":10088},[191],[42,10090],{"className":10091,"style":238},[212],[42,10093,10095,10098,10101,10104,10107],{"className":10094},[133],[42,10096],{"className":10097,"style":4934},[137],[42,10099,4489],{"className":10100},[142,146],[42,10102],{"className":10103,"style":213},[212],[42,10105,4717],{"className":10106},[217],[42,10108],{"className":10109,"style":213},[212],[42,10111,10113,10116,10119,10122],{"className":10112},[133],[42,10114],{"className":10115,"style":708},[137],[42,10117,10046],{"className":10118},[142],[42,10120],{"className":10121},[212,1429],[42,10123],{"className":10124,"style":6047},[212],[42,10126,10128,10131,10134,10143,10146,10149],{"className":10127},[133],[42,10129],{"className":10130,"style":227},[137],[42,10132,92],{"className":10133},[231],[42,10135,10137],{"className":10136},[142],[42,10138,10140],{"className":10139},[142],[42,10141,1451],{"className":10142},[142,1450],[42,10144],{"className":10145,"style":4899},[212],[42,10147,10065],{"className":10148},[142],[42,10150,112],{"className":10151},[316],":",[6251,10154,10155,10438,10726,10909],{},[566,10156,8737,10157,10246,10247,10275,10276,10304,10305,413],{},[42,10158,10160,10192],{"className":10159,"translate":46},[50],[42,10161,10163],{"className":10162},[54],[56,10164,10165],{"xmlns":58},[61,10166,10167,10189],{},[64,10168,10169,10171,10173,10175,10177,10179,10181,10183,10185,10187],{},[70,10170,6601],{},[70,10172,6604],{},[70,10174,1256],{},[79,10176,92],{"stretchy":91},[83,10178,3839],{},[79,10180,3985],{"separator":129},[83,10182,10065],{},[79,10184,112],{"stretchy":91},[79,10186,88],{},[83,10188,85],{},[121,10190,10191],{"encoding":123},"gcd(5,9) = 1",[42,10193,10195,10237],{"className":10194,"ariaHidden":129},[128],[42,10196,10198,10201,10204,10207,10210,10213,10216,10219,10222,10225,10228,10231,10234],{"className":10197},[133],[42,10199],{"className":10200,"style":227},[137],[42,10202,6601],{"className":10203,"style":4535},[142,146],[42,10205,6604],{"className":10206},[142,146],[42,10208,1256],{"className":10209},[142,146],[42,10211,92],{"className":10212},[231],[42,10214,3839],{"className":10215},[142],[42,10217,3985],{"className":10218},[4035],[42,10220],{"className":10221,"style":1455},[212],[42,10223,10065],{"className":10224},[142],[42,10226,112],{"className":10227},[316],[42,10229],{"className":10230,"style":213},[212],[42,10232,88],{"className":10233},[217],[42,10235],{"className":10236,"style":213},[212],[42,10238,10240,10243],{"className":10239},[133],[42,10241],{"className":10242,"style":708},[137],[42,10244,85],{"className":10245},[142],", we know that ",[42,10248,10250,10263],{"className":10249,"translate":46},[50],[42,10251,10253],{"className":10252},[54],[56,10254,10255],{"xmlns":58},[61,10256,10257,10261],{},[64,10258,10259],{},[83,10260,3839],{},[121,10262,3839],{"encoding":123},[42,10264,10266],{"className":10265,"ariaHidden":129},[128],[42,10267,10269,10272],{"className":10268},[133],[42,10270],{"className":10271,"style":708},[137],[42,10273,3839],{"className":10274},[142]," has an inverse modulo ",[42,10277,10279,10292],{"className":10278,"translate":46},[50],[42,10280,10282],{"className":10281},[54],[56,10283,10284],{"xmlns":58},[61,10285,10286,10290],{},[64,10287,10288],{},[83,10289,10065],{},[121,10291,10065],{"encoding":123},[42,10293,10295],{"className":10294,"ariaHidden":129},[128],[42,10296,10298,10301],{"className":10297},[133],[42,10299],{"className":10300,"style":708},[137],[42,10302,10065],{"className":10303},[142],". Using the extended Euclidean algorithm, we find ",[42,10306,10308,10344],{"className":10307,"translate":46},[50],[42,10309,10311],{"className":10310},[54],[56,10312,10313],{"xmlns":58},[61,10314,10315,10341],{},[64,10316,10317,10319,10321,10323,10325,10327,10329,10331,10333,10335,10337,10339],{},[83,10318,1074],{},[79,10320,98],{},[83,10322,3839],{},[79,10324,81],{},[79,10326,92],{"stretchy":91},[79,10328,674],{},[83,10330,85],{},[79,10332,112],{"stretchy":91},[79,10334,98],{},[83,10336,10065],{},[79,10338,88],{},[83,10340,85],{},[121,10342,10343],{"encoding":123},"2*5 + (-1)*9 = 1",[42,10345,10347,10365,10384,10411,10429],{"className":10346,"ariaHidden":129},[128],[42,10348,10350,10353,10356,10359,10362],{"className":10349},[133],[42,10351],{"className":10352,"style":708},[137],[42,10354,1074],{"className":10355},[142],[42,10357],{"className":10358,"style":238},[212],[42,10360,98],{"className":10361},[191],[42,10363],{"className":10364,"style":238},[212],[42,10366,10368,10372,10375,10378,10381],{"className":10367},[133],[42,10369],{"className":10370,"style":10371},[137],"height:0.7278em;vertical-align:-0.0833em;",[42,10373,3839],{"className":10374},[142],[42,10376],{"className":10377,"style":238},[212],[42,10379,81],{"className":10380},[191],[42,10382],{"className":10383,"style":238},[212],[42,10385,10387,10390,10393,10396,10399,10402,10405,10408],{"className":10386},[133],[42,10388],{"className":10389,"style":227},[137],[42,10391,92],{"className":10392},[231],[42,10394,674],{"className":10395},[142],[42,10397,85],{"className":10398},[142],[42,10400,112],{"className":10401},[316],[42,10403],{"className":10404,"style":238},[212],[42,10406,98],{"className":10407},[191],[42,10409],{"className":10410,"style":238},[212],[42,10412,10414,10417,10420,10423,10426],{"className":10413},[133],[42,10415],{"className":10416,"style":708},[137],[42,10418,10065],{"className":10419},[142],[42,10421],{"className":10422,"style":213},[212],[42,10424,88],{"className":10425},[217],[42,10427],{"className":10428,"style":213},[212],[42,10430,10432,10435],{"className":10431},[133],[42,10433],{"className":10434,"style":708},[137],[42,10436,85],{"className":10437},[142],[566,10439,10440,10441,10469,10470,8703,10498,10526,10527,10555,10556,413],{},"Therefore, ",[42,10442,10444,10457],{"className":10443,"translate":46},[50],[42,10445,10447],{"className":10446},[54],[56,10448,10449],{"xmlns":58},[61,10450,10451,10455],{},[64,10452,10453],{},[83,10454,1074],{},[121,10456,1074],{"encoding":123},[42,10458,10460],{"className":10459,"ariaHidden":129},[128],[42,10461,10463,10466],{"className":10462},[133],[42,10464],{"className":10465,"style":708},[137],[42,10467,1074],{"className":10468},[142]," is the inverse of ",[42,10471,10473,10486],{"className":10472,"translate":46},[50],[42,10474,10476],{"className":10475},[54],[56,10477,10478],{"xmlns":58},[61,10479,10480,10484],{},[64,10481,10482],{},[83,10483,3839],{},[121,10485,3839],{"encoding":123},[42,10487,10489],{"className":10488,"ariaHidden":129},[128],[42,10490,10492,10495],{"className":10491},[133],[42,10493],{"className":10494,"style":708},[137],[42,10496,3839],{"className":10497},[142],[42,10499,10501,10514],{"className":10500,"translate":46},[50],[42,10502,10504],{"className":10503},[54],[56,10505,10506],{"xmlns":58},[61,10507,10508,10512],{},[64,10509,10510],{},[83,10511,10065],{},[121,10513,10065],{"encoding":123},[42,10515,10517],{"className":10516,"ariaHidden":129},[128],[42,10518,10520,10523],{"className":10519},[133],[42,10521],{"className":10522,"style":708},[137],[42,10524,10065],{"className":10525},[142],". Multiplying both sides of the original equation by ",[42,10528,10530,10543],{"className":10529,"translate":46},[50],[42,10531,10533],{"className":10532},[54],[56,10534,10535],{"xmlns":58},[61,10536,10537,10541],{},[64,10538,10539],{},[83,10540,1074],{},[121,10542,1074],{"encoding":123},[42,10544,10546],{"className":10545,"ariaHidden":129},[128],[42,10547,10549,10552],{"className":10548},[133],[42,10550],{"className":10551,"style":708},[137],[42,10553,1074],{"className":10554},[142]," gives ",[42,10557,10559,10609],{"className":10558,"translate":46},[50],[42,10560,10562],{"className":10561},[54],[56,10563,10564],{"xmlns":58},[61,10565,10566,10606],{},[64,10567,10568,10570,10572,10574,10576,10578,10580,10582,10584,10586,10588,10590,10592,10600,10602,10604],{},[83,10569,1074],{},[79,10571,98],{},[83,10573,3839],{},[79,10575,98],{},[70,10577,4489],{},[79,10579,4717],{},[83,10581,1074],{},[79,10583,98],{},[83,10585,10046],{},[212,10587],{},[212,10589],{"width":5973},[79,10591,92],{"stretchy":91},[64,10593,10594,10596,10598],{},[70,10595,1250],{"mathvariant":115},[70,10597,1253],{"mathvariant":115},[70,10599,1256],{"mathvariant":115},[212,10601],{"width":4749},[83,10603,10065],{},[79,10605,112],{"stretchy":91},[121,10607,10608],{"encoding":123},"2*5*x ≡ 2*7 \\pmod{9}",[42,10610,10612,10630,10648,10666,10684,10699],{"className":10611,"ariaHidden":129},[128],[42,10613,10615,10618,10621,10624,10627],{"className":10614},[133],[42,10616],{"className":10617,"style":708},[137],[42,10619,1074],{"className":10620},[142],[42,10622],{"className":10623,"style":238},[212],[42,10625,98],{"className":10626},[191],[42,10628],{"className":10629,"style":238},[212],[42,10631,10633,10636,10639,10642,10645],{"className":10632},[133],[42,10634],{"className":10635,"style":708},[137],[42,10637,3839],{"className":10638},[142],[42,10640],{"className":10641,"style":238},[212],[42,10643,98],{"className":10644},[191],[42,10646],{"className":10647,"style":238},[212],[42,10649,10651,10654,10657,10660,10663],{"className":10650},[133],[42,10652],{"className":10653,"style":4934},[137],[42,10655,4489],{"className":10656},[142,146],[42,10658],{"className":10659,"style":213},[212],[42,10661,4717],{"className":10662},[217],[42,10664],{"className":10665,"style":213},[212],[42,10667,10669,10672,10675,10678,10681],{"className":10668},[133],[42,10670],{"className":10671,"style":708},[137],[42,10673,1074],{"className":10674},[142],[42,10676],{"className":10677,"style":238},[212],[42,10679,98],{"className":10680},[191],[42,10682],{"className":10683,"style":238},[212],[42,10685,10687,10690,10693,10696],{"className":10686},[133],[42,10688],{"className":10689,"style":708},[137],[42,10691,10046],{"className":10692},[142],[42,10694],{"className":10695},[212,1429],[42,10697],{"className":10698,"style":6047},[212],[42,10700,10702,10705,10708,10717,10720,10723],{"className":10701},[133],[42,10703],{"className":10704,"style":227},[137],[42,10706,92],{"className":10707},[231],[42,10709,10711],{"className":10710},[142],[42,10712,10714],{"className":10713},[142],[42,10715,1451],{"className":10716},[142,1450],[42,10718],{"className":10719,"style":4899},[212],[42,10721,10065],{"className":10722},[142],[42,10724,112],{"className":10725},[316],[566,10727,10728,10729,413],{},"Simplifying, we get ",[42,10730,10732,10795],{"className":10731,"translate":46},[50],[42,10733,10735],{"className":10734},[54],[56,10736,10737],{"xmlns":58},[61,10738,10739,10792],{},[64,10740,10741,10743,10745,10748,10750,10752,10754,10762,10764,10766,10768,10770,10772,10774,10776,10778,10786,10788,10790],{},[70,10742,4489],{},[79,10744,4717],{},[83,10746,10747],{},"14",[212,10749],{},[212,10751],{"width":5973},[79,10753,92],{"stretchy":91},[64,10755,10756,10758,10760],{},[70,10757,1250],{"mathvariant":115},[70,10759,1253],{"mathvariant":115},[70,10761,1256],{"mathvariant":115},[212,10763],{"width":4749},[83,10765,10065],{},[79,10767,112],{"stretchy":91},[79,10769,4717],{},[83,10771,3839],{},[212,10773],{},[212,10775],{"width":5973},[79,10777,92],{"stretchy":91},[64,10779,10780,10782,10784],{},[70,10781,1250],{"mathvariant":115},[70,10783,1253],{"mathvariant":115},[70,10785,1256],{"mathvariant":115},[212,10787],{"width":4749},[83,10789,10065],{},[79,10791,112],{"stretchy":91},[121,10793,10794],{"encoding":123},"x ≡ 14 \\pmod{9} ≡ 5 \\pmod{9}",[42,10796,10798,10816,10831,10867,10882],{"className":10797,"ariaHidden":129},[128],[42,10799,10801,10804,10807,10810,10813],{"className":10800},[133],[42,10802],{"className":10803,"style":4934},[137],[42,10805,4489],{"className":10806},[142,146],[42,10808],{"className":10809,"style":213},[212],[42,10811,4717],{"className":10812},[217],[42,10814],{"className":10815,"style":213},[212],[42,10817,10819,10822,10825,10828],{"className":10818},[133],[42,10820],{"className":10821,"style":708},[137],[42,10823,10747],{"className":10824},[142],[42,10826],{"className":10827},[212,1429],[42,10829],{"className":10830,"style":6047},[212],[42,10832,10834,10837,10840,10849,10852,10855,10858,10861,10864],{"className":10833},[133],[42,10835],{"className":10836,"style":227},[137],[42,10838,92],{"className":10839},[231],[42,10841,10843],{"className":10842},[142],[42,10844,10846],{"className":10845},[142],[42,10847,1451],{"className":10848},[142,1450],[42,10850],{"className":10851,"style":4899},[212],[42,10853,10065],{"className":10854},[142],[42,10856,112],{"className":10857},[316],[42,10859],{"className":10860,"style":213},[212],[42,10862,4717],{"className":10863},[217],[42,10865],{"className":10866,"style":213},[212],[42,10868,10870,10873,10876,10879],{"className":10869},[133],[42,10871],{"className":10872,"style":708},[137],[42,10874,3839],{"className":10875},[142],[42,10877],{"className":10878},[212,1429],[42,10880],{"className":10881,"style":6047},[212],[42,10883,10885,10888,10891,10900,10903,10906],{"className":10884},[133],[42,10886],{"className":10887,"style":227},[137],[42,10889,92],{"className":10890},[231],[42,10892,10894],{"className":10893},[142],[42,10895,10897],{"className":10896},[142],[42,10898,1451],{"className":10899},[142,1450],[42,10901],{"className":10902,"style":4899},[212],[42,10904,10065],{"className":10905},[142],[42,10907,112],{"className":10908},[316],[566,10910,10911,10912,413],{},"Thus, the solution is ",[42,10913,10915,10939],{"className":10914,"translate":46},[50],[42,10916,10918],{"className":10917},[54],[56,10919,10920],{"xmlns":58},[61,10921,10922,10936],{},[64,10923,10924,10926,10928,10930,10932,10934],{},[70,10925,4489],{},[79,10927,88],{},[83,10929,3839],{},[79,10931,81],{},[83,10933,10065],{},[70,10935,1513],{},[121,10937,10938],{"encoding":123},"x = 5 + 9k",[42,10940,10942,10960,10978],{"className":10941,"ariaHidden":129},[128],[42,10943,10945,10948,10951,10954,10957],{"className":10944},[133],[42,10946],{"className":10947,"style":4501},[137],[42,10949,4489],{"className":10950},[142,146],[42,10952],{"className":10953,"style":213},[212],[42,10955,88],{"className":10956},[217],[42,10958],{"className":10959,"style":213},[212],[42,10961,10963,10966,10969,10972,10975],{"className":10962},[133],[42,10964],{"className":10965,"style":10371},[137],[42,10967,3839],{"className":10968},[142],[42,10970],{"className":10971,"style":238},[212],[42,10973,81],{"className":10974},[191],[42,10976],{"className":10977,"style":238},[212],[42,10979,10981,10984,10987],{"className":10980},[133],[42,10982],{"className":10983,"style":1440},[137],[42,10985,10065],{"className":10986},[142],[42,10988,1513],{"className":10989,"style":1572},[142,146],[15,10991,10992,10993,11119,11120,11368,11369,11581,11582,413],{},"In summary, we can solve a general linear congruence ",[42,10994,10996,11038],{"className":10995,"translate":46},[50],[42,10997,10999],{"className":10998},[54],[56,11000,11001],{"xmlns":58},[61,11002,11003,11035],{},[64,11004,11005,11007,11009,11011,11013,11015,11017,11019,11021,11029,11031,11033],{},[70,11006,2626],{},[79,11008,98],{},[70,11010,4489],{},[79,11012,4717],{},[70,11014,5700],{},[212,11016],{},[212,11018],{"width":5973},[79,11020,92],{"stretchy":91},[64,11022,11023,11025,11027],{},[70,11024,1250],{"mathvariant":115},[70,11026,1253],{"mathvariant":115},[70,11028,1256],{"mathvariant":115},[212,11030],{"width":4749},[70,11032,4667],{},[79,11034,112],{"stretchy":91},[121,11036,11037],{"encoding":123},"a*x ≡ b \\pmod{L}",[42,11039,11041,11059,11077,11092],{"className":11040,"ariaHidden":129},[128],[42,11042,11044,11047,11050,11053,11056],{"className":11043},[133],[42,11045],{"className":11046,"style":2661},[137],[42,11048,2626],{"className":11049},[142,146],[42,11051],{"className":11052,"style":238},[212],[42,11054,98],{"className":11055},[191],[42,11057],{"className":11058,"style":238},[212],[42,11060,11062,11065,11068,11071,11074],{"className":11061},[133],[42,11063],{"className":11064,"style":4934},[137],[42,11066,4489],{"className":11067},[142,146],[42,11069],{"className":11070,"style":213},[212],[42,11072,4717],{"className":11073},[217],[42,11075],{"className":11076,"style":213},[212],[42,11078,11080,11083,11086,11089],{"className":11079},[133],[42,11081],{"className":11082,"style":1440},[137],[42,11084,5700],{"className":11085},[142,146],[42,11087],{"className":11088},[212,1429],[42,11090],{"className":11091,"style":6047},[212],[42,11093,11095,11098,11101,11110,11113,11116],{"className":11094},[133],[42,11096],{"className":11097,"style":227},[137],[42,11099,92],{"className":11100},[231],[42,11102,11104],{"className":11103},[142],[42,11105,11107],{"className":11106},[142],[42,11108,1451],{"className":11109},[142,1450],[42,11111],{"className":11112,"style":4899},[212],[42,11114,4667],{"className":11115},[142,146],[42,11117,112],{"className":11118},[316]," by transforming it into a standard congruence ",[42,11121,11123,11176],{"className":11122,"translate":46},[50],[42,11124,11126],{"className":11125},[54],[56,11127,11128],{"xmlns":58},[61,11129,11130,11174],{},[64,11131,11132,11138,11140,11146,11148,11150,11152,11154,11156,11164,11166,11172],{},[67,11133,11134,11136],{},[70,11135,2626],{},[83,11137,739],{},[79,11139,98],{},[67,11141,11142,11144],{},[70,11143,4489],{},[83,11145,739],{},[79,11147,4717],{},[83,11149,85],{},[212,11151],{},[212,11153],{"width":5973},[79,11155,92],{"stretchy":91},[64,11157,11158,11160,11162],{},[70,11159,1250],{"mathvariant":115},[70,11161,1253],{"mathvariant":115},[70,11163,1256],{"mathvariant":115},[212,11165],{"width":4749},[67,11167,11168,11170],{},[70,11169,119],{},[83,11171,739],{},[79,11173,112],{"stretchy":91},[121,11175,8290],{"encoding":123},[42,11177,11179,11234,11289,11304],{"className":11178,"ariaHidden":129},[128],[42,11180,11182,11185,11225,11228,11231],{"className":11181},[133],[42,11183],{"className":11184,"style":6791},[137],[42,11186,11188,11191],{"className":11187},[142],[42,11189,2626],{"className":11190},[142,146],[42,11192,11194],{"className":11193},[151],[42,11195,11197,11217],{"className":11196},[155,156],[42,11198,11200,11214],{"className":11199},[160],[42,11201,11203],{"className":11202,"style":937},[164],[42,11204,11205,11208],{"style":6812},[42,11206],{"className":11207,"style":173},[172],[42,11209,11211],{"className":11210},[177,178,179,180],[42,11212,739],{"className":11213},[142,180],[42,11215,199],{"className":11216},[198],[42,11218,11220],{"className":11219},[160],[42,11221,11223],{"className":11222,"style":961},[164],[42,11224],{},[42,11226],{"className":11227,"style":238},[212],[42,11229,98],{"className":11230},[191],[42,11232],{"className":11233,"style":238},[212],[42,11235,11237,11240,11280,11283,11286],{"className":11236},[133],[42,11238],{"className":11239,"style":8355},[137],[42,11241,11243,11246],{"className":11242},[142],[42,11244,4489],{"className":11245},[142,146],[42,11247,11249],{"className":11248},[151],[42,11250,11252,11272],{"className":11251},[155,156],[42,11253,11255,11269],{"className":11254},[160],[42,11256,11258],{"className":11257,"style":937},[164],[42,11259,11260,11263],{"style":6812},[42,11261],{"className":11262,"style":173},[172],[42,11264,11266],{"className":11265},[177,178,179,180],[42,11267,739],{"className":11268},[142,180],[42,11270,199],{"className":11271},[198],[42,11273,11275],{"className":11274},[160],[42,11276,11278],{"className":11277,"style":961},[164],[42,11279],{},[42,11281],{"className":11282,"style":213},[212],[42,11284,4717],{"className":11285},[217],[42,11287],{"className":11288,"style":213},[212],[42,11290,11292,11295,11298,11301],{"className":11291},[133],[42,11293],{"className":11294,"style":708},[137],[42,11296,85],{"className":11297},[142],[42,11299],{"className":11300},[212,1429],[42,11302],{"className":11303,"style":6047},[212],[42,11305,11307,11310,11313,11322,11325,11365],{"className":11306},[133],[42,11308],{"className":11309,"style":227},[137],[42,11311,92],{"className":11312},[231],[42,11314,11316],{"className":11315},[142],[42,11317,11319],{"className":11318},[142],[42,11320,1451],{"className":11321},[142,1450],[42,11323],{"className":11324,"style":4899},[212],[42,11326,11328,11331],{"className":11327},[142],[42,11329,119],{"className":11330,"style":147},[142,146],[42,11332,11334],{"className":11333},[151],[42,11335,11337,11357],{"className":11336},[155,156],[42,11338,11340,11354],{"className":11339},[160],[42,11341,11343],{"className":11342,"style":937},[164],[42,11344,11345,11348],{"style":168},[42,11346],{"className":11347,"style":173},[172],[42,11349,11351],{"className":11350},[177,178,179,180],[42,11352,739],{"className":11353},[142,180],[42,11355,199],{"className":11356},[198],[42,11358,11360],{"className":11359},[160],[42,11361,11363],{"className":11362,"style":961},[164],[42,11364],{},[42,11366,112],{"className":11367},[316],". After solving the standard congruence, we map the solution back. If the solution to the standard equation is ",[42,11370,11372,11416],{"className":11371,"translate":46},[50],[42,11373,11375],{"className":11374},[54],[56,11376,11377],{"xmlns":58},[61,11378,11379,11413],{},[64,11380,11381,11387,11390,11392,11394,11396,11398,11400,11403,11405,11411],{},[67,11382,11383,11385],{},[70,11384,4489],{},[83,11386,739],{},[79,11388,11389],{},"∈",[79,11391,3980],{"stretchy":91},[70,11393,4489],{},[79,11395,88],{},[70,11397,15],{},[79,11399,81],{},[70,11401,11402],{},"q",[79,11404,98],{},[67,11406,11407,11409],{},[70,11408,119],{},[83,11410,739],{},[79,11412,4013],{"stretchy":91},[121,11414,11415],{"encoding":123},"x_0 \\in \\{x = p + q*M_0\\}",[42,11417,11419,11475,11496,11514,11532],{"className":11418,"ariaHidden":129},[128],[42,11420,11422,11426,11466,11469,11472],{"className":11421},[133],[42,11423],{"className":11424,"style":11425},[137],"height:0.6891em;vertical-align:-0.15em;",[42,11427,11429,11432],{"className":11428},[142],[42,11430,4489],{"className":11431},[142,146],[42,11433,11435],{"className":11434},[151],[42,11436,11438,11458],{"className":11437},[155,156],[42,11439,11441,11455],{"className":11440},[160],[42,11442,11444],{"className":11443,"style":937},[164],[42,11445,11446,11449],{"style":6812},[42,11447],{"className":11448,"style":173},[172],[42,11450,11452],{"className":11451},[177,178,179,180],[42,11453,739],{"className":11454},[142,180],[42,11456,199],{"className":11457},[198],[42,11459,11461],{"className":11460},[160],[42,11462,11464],{"className":11463,"style":961},[164],[42,11465],{},[42,11467],{"className":11468,"style":213},[212],[42,11470,11389],{"className":11471},[217],[42,11473],{"className":11474,"style":213},[212],[42,11476,11478,11481,11484,11487,11490,11493],{"className":11477},[133],[42,11479],{"className":11480,"style":227},[137],[42,11482,3980],{"className":11483},[231],[42,11485,4489],{"className":11486},[142,146],[42,11488],{"className":11489,"style":213},[212],[42,11491,88],{"className":11492},[217],[42,11494],{"className":11495,"style":213},[212],[42,11497,11499,11502,11505,11508,11511],{"className":11498},[133],[42,11500],{"className":11501,"style":5088},[137],[42,11503,15],{"className":11504},[142,146],[42,11506],{"className":11507,"style":238},[212],[42,11509,81],{"className":11510},[191],[42,11512],{"className":11513,"style":238},[212],[42,11515,11517,11520,11523,11526,11529],{"className":11516},[133],[42,11518],{"className":11519,"style":7853},[137],[42,11521,11402],{"className":11522,"style":4535},[142,146],[42,11524],{"className":11525,"style":238},[212],[42,11527,98],{"className":11528},[191],[42,11530],{"className":11531,"style":238},[212],[42,11533,11535,11538,11578],{"className":11534},[133],[42,11536],{"className":11537,"style":227},[137],[42,11539,11541,11544],{"className":11540},[142],[42,11542,119],{"className":11543,"style":147},[142,146],[42,11545,11547],{"className":11546},[151],[42,11548,11550,11570],{"className":11549},[155,156],[42,11551,11553,11567],{"className":11552},[160],[42,11554,11556],{"className":11555,"style":937},[164],[42,11557,11558,11561],{"style":168},[42,11559],{"className":11560,"style":173},[172],[42,11562,11564],{"className":11563},[177,178,179,180],[42,11565,739],{"className":11566},[142,180],[42,11568,199],{"className":11569},[198],[42,11571,11573],{"className":11572},[160],[42,11574,11576],{"className":11575,"style":961},[164],[42,11577],{},[42,11579,4013],{"className":11580},[316],", then the solution to the original equation is ",[42,11583,11585,11636],{"className":11584,"translate":46},[50],[42,11586,11588],{"className":11587},[54],[56,11589,11590],{"xmlns":58},[61,11591,11592,11633],{},[64,11593,11594,11596,11598,11600,11602,11604,11611,11613,11615,11617,11623,11625,11631],{},[70,11595,4489],{},[79,11597,11389],{},[79,11599,3980],{"stretchy":91},[70,11601,4489],{},[79,11603,88],{},[11605,11606,11607,11609],"mfrac",{},[70,11608,5700],{},[70,11610,1256],{},[79,11612,98],{},[70,11614,15],{},[79,11616,81],{},[1070,11618,11619,11621],{},[70,11620,11402],{},[79,11622,5020],{"mathvariant":115,"lspace":5019,"rspace":5019},[79,11624,98],{},[67,11626,11627,11629],{},[70,11628,119],{},[83,11630,739],{},[79,11632,4013],{"stretchy":91},[121,11634,11635],{"encoding":123},"x \\in \\{x = \\frac{b}{d}*p + q'*M_0\\}",[42,11637,11639,11658,11679,11771,11789,11838],{"className":11638,"ariaHidden":129},[128],[42,11640,11642,11646,11649,11652,11655],{"className":11641},[133],[42,11643],{"className":11644,"style":11645},[137],"height:0.5782em;vertical-align:-0.0391em;",[42,11647,4489],{"className":11648},[142,146],[42,11650],{"className":11651,"style":213},[212],[42,11653,11389],{"className":11654},[217],[42,11656],{"className":11657,"style":213},[212],[42,11659,11661,11664,11667,11670,11673,11676],{"className":11660},[133],[42,11662],{"className":11663,"style":227},[137],[42,11665,3980],{"className":11666},[231],[42,11668,4489],{"className":11669},[142,146],[42,11671],{"className":11672,"style":213},[212],[42,11674,88],{"className":11675},[217],[42,11677],{"className":11678,"style":213},[212],[42,11680,11682,11686,11762,11765,11768],{"className":11681},[133],[42,11683],{"className":11684,"style":11685},[137],"height:1.2251em;vertical-align:-0.345em;",[42,11687,11689,11693,11759],{"className":11688},[142],[42,11690],{"className":11691},[231,11692],"nulldelimiter",[42,11694,11696],{"className":11695},[11605],[42,11697,11699,11750],{"className":11698},[155,156],[42,11700,11702,11747],{"className":11701},[160],[42,11703,11706,11721,11732],{"className":11704,"style":11705},[164],"height:0.8801em;",[42,11707,11709,11712],{"style":11708},"top:-2.655em;",[42,11710],{"className":11711,"style":9515},[172],[42,11713,11715],{"className":11714},[177,178,179,180],[42,11716,11718],{"className":11717},[142,180],[42,11719,1256],{"className":11720},[142,146,180],[42,11722,11724,11727],{"style":11723},"top:-3.23em;",[42,11725],{"className":11726,"style":9515},[172],[42,11728],{"className":11729,"style":11731},[11730],"frac-line","border-bottom-width:0.04em;",[42,11733,11735,11738],{"style":11734},"top:-3.394em;",[42,11736],{"className":11737,"style":9515},[172],[42,11739,11741],{"className":11740},[177,178,179,180],[42,11742,11744],{"className":11743},[142,180],[42,11745,5700],{"className":11746},[142,146,180],[42,11748,199],{"className":11749},[198],[42,11751,11753],{"className":11752},[160],[42,11754,11757],{"className":11755,"style":11756},[164],"height:0.345em;",[42,11758],{},[42,11760],{"className":11761},[316,11692],[42,11763],{"className":11764,"style":238},[212],[42,11766,98],{"className":11767},[191],[42,11769],{"className":11770,"style":238},[212],[42,11772,11774,11777,11780,11783,11786],{"className":11773},[133],[42,11775],{"className":11776,"style":5088},[137],[42,11778,15],{"className":11779},[142,146],[42,11781],{"className":11782,"style":238},[212],[42,11784,81],{"className":11785},[191],[42,11787],{"className":11788,"style":238},[212],[42,11790,11792,11796,11829,11832,11835],{"className":11791},[133],[42,11793],{"className":11794,"style":11795},[137],"height:0.9463em;vertical-align:-0.1944em;",[42,11797,11799,11802],{"className":11798},[142],[42,11800,11402],{"className":11801,"style":4535},[142,146],[42,11803,11805],{"className":11804},[151],[42,11806,11808],{"className":11807},[155],[42,11809,11811],{"className":11810},[160],[42,11812,11815],{"className":11813,"style":11814},[164],"height:0.7519em;",[42,11816,11817,11820],{"style":1111},[42,11818],{"className":11819,"style":173},[172],[42,11821,11823],{"className":11822},[177,178,179,180],[42,11824,11826],{"className":11825},[142,180],[42,11827,5020],{"className":11828},[142,180],[42,11830],{"className":11831,"style":238},[212],[42,11833,98],{"className":11834},[191],[42,11836],{"className":11837,"style":238},[212],[42,11839,11841,11844,11884],{"className":11840},[133],[42,11842],{"className":11843,"style":227},[137],[42,11845,11847,11850],{"className":11846},[142],[42,11848,119],{"className":11849,"style":147},[142,146],[42,11851,11853],{"className":11852},[151],[42,11854,11856,11876],{"className":11855},[155,156],[42,11857,11859,11873],{"className":11858},[160],[42,11860,11862],{"className":11861,"style":937},[164],[42,11863,11864,11867],{"style":168},[42,11865],{"className":11866,"style":173},[172],[42,11868,11870],{"className":11869},[177,178,179,180],[42,11871,739],{"className":11872},[142,180],[42,11874,199],{"className":11875},[198],[42,11877,11879],{"className":11878},[160],[42,11880,11882],{"className":11881,"style":961},[164],[42,11883],{},[42,11885,4013],{"className":11886},[316],[15,11888,11889],{},"Through this, we have outlined the process of solving linear congruences and deepened our understanding of the linear congruential method.",[10,11891,11893],{"id":11892},"insecurity-of-the-linear-congruential-method","Insecurity of the Linear Congruential Method",[15,11895,11896],{},"The linear congruential method (LCG) is a pseudo-random number generation technique based on deterministic mathematical algorithms. Although it can meet the needs for general pseudo-random numbers, it is not suitable for high-security applications, such as cryptography or applications requiring strong randomness. Below are some key considerations regarding the insecurity of LCG:",[6251,11898,11899,11905,11911,11917],{},[566,11900,11901,11904],{},[24,11902,11903],{},"Determinism",": LCG is deterministic, meaning that the same seed and parameters will always produce the same sequence of random numbers. This poses a security risk, as an attacker could reproduce the sequence by guessing the seed and parameters, breaking the randomness.",[566,11906,11907,11910],{},[24,11908,11909],{},"Periodicity",": The random number sequences generated by LCG are periodic. Once the sequence reaches its period, it starts repeating. This periodicity can cause problems in applications that require long-term randomness.",[566,11912,11913,11916],{},[24,11914,11915],{},"Non-uniformity",": In some cases, LCG-generated sequences may not be uniformly distributed, especially when parameters are poorly chosen, leading to statistical biases.",[566,11918,11919,11922],{},[24,11920,11921],{},"Security",": LCG is unsuitable for cryptographic or high-randomness security applications. For such use cases, cryptographically secure pseudo-random number generators (CSPRNGs) must be employed.",[15,11924,11925],{},"In conclusion, while LCG can be used for general pseudo-random number generation in simulations, games, or random testing, it should not be used in applications that require high security and strong randomness.",[10,11927,11929],{"id":11928},"code-implementation","Code Implementation",[15,11931,11932],{},"Below is the implementation of a Linear Congruential Generator (LCG) using TypeScript, along with a simple program to test the effectiveness of its randomness.",[11934,11935,11937],"h3",{"id":11936},"lcg-implementation","LCG Implementation",[11939,11940,11945],"pre",{"className":11941,"code":11942,"language":11943,"meta":11944,"style":11944},"language-typescript shiki shiki-themes github-dark","class LinearCongruentialGenerator {\n  private seed: number;\n  private readonly a: number;\n  private readonly c: number;\n  private readonly m: number;\n\n  constructor(\n    seed: number,\n    a: number = 1664525,\n    c: number = 1013904223,\n    m: number = 2 ** 32\n  ) {\n    this.seed = seed;\n    this.a = a;\n    this.c = c;\n    this.m = m;\n  }\n\n  public next(): number {\n    this.seed = (this.a * this.seed + this.c) % this.m;\n    return this.seed;\n  }\n\n  public nextFloat(): number {\n    return this.next() \u002F this.m;\n  }\n}\n\n\u002F\u002F Example usage\nconst lcg = new LinearCongruentialGenerator(12345);\nconsole.log(lcg.next());\n\u002F\u002F Outputs a random integer\nconsole.log(lcg.nextFloat());\n\u002F\u002F Outputs a random floating-point number between 0 and 1\n","typescript","",[11946,11947,11948,11964,11983,12000,12016,12032,12039,12048,12061,12079,12096,12117,12123,12137,12150,12163,12176,12182,12187,12205,12244,12255,12260,12265,12281,12302,12307,12313,12318,12325,12349,12366,12372,12386],"code",{"__ignoreMap":11944},[42,11949,11952,11956,11960],{"class":11950,"line":11951},"line",1,[42,11953,11955],{"class":11954},"snl16","class",[42,11957,11959],{"class":11958},"svObZ"," LinearCongruentialGenerator",[42,11961,11963],{"class":11962},"s95oV"," {\n",[42,11965,11967,11970,11974,11976,11980],{"class":11950,"line":11966},2,[42,11968,11969],{"class":11954},"  private",[42,11971,11973],{"class":11972},"s9osk"," seed",[42,11975,10152],{"class":11954},[42,11977,11979],{"class":11978},"sDLfK"," number",[42,11981,11982],{"class":11962},";\n",[42,11984,11986,11988,11991,11994,11996,11998],{"class":11950,"line":11985},3,[42,11987,11969],{"class":11954},[42,11989,11990],{"class":11954}," readonly",[42,11992,11993],{"class":11972}," a",[42,11995,10152],{"class":11954},[42,11997,11979],{"class":11978},[42,11999,11982],{"class":11962},[42,12001,12003,12005,12007,12010,12012,12014],{"class":11950,"line":12002},4,[42,12004,11969],{"class":11954},[42,12006,11990],{"class":11954},[42,12008,12009],{"class":11972}," c",[42,12011,10152],{"class":11954},[42,12013,11979],{"class":11978},[42,12015,11982],{"class":11962},[42,12017,12019,12021,12023,12026,12028,12030],{"class":11950,"line":12018},5,[42,12020,11969],{"class":11954},[42,12022,11990],{"class":11954},[42,12024,12025],{"class":11972}," m",[42,12027,10152],{"class":11954},[42,12029,11979],{"class":11978},[42,12031,11982],{"class":11962},[42,12033,12035],{"class":11950,"line":12034},6,[42,12036,12038],{"emptyLinePlaceholder":12037},true,"\n",[42,12040,12042,12045],{"class":11950,"line":12041},7,[42,12043,12044],{"class":11954},"  constructor",[42,12046,12047],{"class":11962},"(\n",[42,12049,12051,12054,12056,12058],{"class":11950,"line":12050},8,[42,12052,12053],{"class":11972},"    seed",[42,12055,10152],{"class":11954},[42,12057,11979],{"class":11978},[42,12059,12060],{"class":11962},",\n",[42,12062,12064,12067,12069,12071,12074,12077],{"class":11950,"line":12063},9,[42,12065,12066],{"class":11972},"    a",[42,12068,10152],{"class":11954},[42,12070,11979],{"class":11978},[42,12072,12073],{"class":11954}," =",[42,12075,12076],{"class":11978}," 1664525",[42,12078,12060],{"class":11962},[42,12080,12082,12085,12087,12089,12091,12094],{"class":11950,"line":12081},10,[42,12083,12084],{"class":11972},"    c",[42,12086,10152],{"class":11954},[42,12088,11979],{"class":11978},[42,12090,12073],{"class":11954},[42,12092,12093],{"class":11978}," 1013904223",[42,12095,12060],{"class":11962},[42,12097,12099,12102,12104,12106,12108,12111,12114],{"class":11950,"line":12098},11,[42,12100,12101],{"class":11972},"    m",[42,12103,10152],{"class":11954},[42,12105,11979],{"class":11978},[42,12107,12073],{"class":11954},[42,12109,12110],{"class":11978}," 2",[42,12112,12113],{"class":11954}," **",[42,12115,12116],{"class":11978}," 32\n",[42,12118,12120],{"class":11950,"line":12119},12,[42,12121,12122],{"class":11962},"  ) {\n",[42,12124,12126,12129,12132,12134],{"class":11950,"line":12125},13,[42,12127,12128],{"class":11978},"    this",[42,12130,12131],{"class":11962},".seed ",[42,12133,88],{"class":11954},[42,12135,12136],{"class":11962}," seed;\n",[42,12138,12140,12142,12145,12147],{"class":11950,"line":12139},14,[42,12141,12128],{"class":11978},[42,12143,12144],{"class":11962},".a ",[42,12146,88],{"class":11954},[42,12148,12149],{"class":11962}," a;\n",[42,12151,12153,12155,12158,12160],{"class":11950,"line":12152},15,[42,12154,12128],{"class":11978},[42,12156,12157],{"class":11962},".c ",[42,12159,88],{"class":11954},[42,12161,12162],{"class":11962}," c;\n",[42,12164,12166,12168,12171,12173],{"class":11950,"line":12165},16,[42,12167,12128],{"class":11978},[42,12169,12170],{"class":11962},".m ",[42,12172,88],{"class":11954},[42,12174,12175],{"class":11962}," m;\n",[42,12177,12179],{"class":11950,"line":12178},17,[42,12180,12181],{"class":11962},"  }\n",[42,12183,12185],{"class":11950,"line":12184},18,[42,12186,12038],{"emptyLinePlaceholder":12037},[42,12188,12190,12193,12196,12199,12201,12203],{"class":11950,"line":12189},19,[42,12191,12192],{"class":11954},"  public",[42,12194,12195],{"class":11958}," next",[42,12197,12198],{"class":11962},"()",[42,12200,10152],{"class":11954},[42,12202,11979],{"class":11978},[42,12204,11963],{"class":11962},[42,12206,12208,12210,12212,12214,12217,12220,12222,12225,12228,12230,12232,12234,12237,12239,12241],{"class":11950,"line":12207},20,[42,12209,12128],{"class":11978},[42,12211,12131],{"class":11962},[42,12213,88],{"class":11954},[42,12215,12216],{"class":11962}," (",[42,12218,12219],{"class":11978},"this",[42,12221,12144],{"class":11962},[42,12223,12224],{"class":11954},"*",[42,12226,12227],{"class":11978}," this",[42,12229,12131],{"class":11962},[42,12231,81],{"class":11954},[42,12233,12227],{"class":11978},[42,12235,12236],{"class":11962},".c) ",[42,12238,116],{"class":11954},[42,12240,12227],{"class":11978},[42,12242,12243],{"class":11962},".m;\n",[42,12245,12247,12250,12252],{"class":11950,"line":12246},21,[42,12248,12249],{"class":11954},"    return",[42,12251,12227],{"class":11978},[42,12253,12254],{"class":11962},".seed;\n",[42,12256,12258],{"class":11950,"line":12257},22,[42,12259,12181],{"class":11962},[42,12261,12263],{"class":11950,"line":12262},23,[42,12264,12038],{"emptyLinePlaceholder":12037},[42,12266,12268,12270,12273,12275,12277,12279],{"class":11950,"line":12267},24,[42,12269,12192],{"class":11954},[42,12271,12272],{"class":11958}," nextFloat",[42,12274,12198],{"class":11962},[42,12276,10152],{"class":11954},[42,12278,11979],{"class":11978},[42,12280,11963],{"class":11962},[42,12282,12284,12286,12288,12290,12293,12296,12298,12300],{"class":11950,"line":12283},25,[42,12285,12249],{"class":11954},[42,12287,12227],{"class":11978},[42,12289,413],{"class":11962},[42,12291,12292],{"class":11958},"next",[42,12294,12295],{"class":11962},"() ",[42,12297,3725],{"class":11954},[42,12299,12227],{"class":11978},[42,12301,12243],{"class":11962},[42,12303,12305],{"class":11950,"line":12304},26,[42,12306,12181],{"class":11962},[42,12308,12310],{"class":11950,"line":12309},27,[42,12311,12312],{"class":11962},"}\n",[42,12314,12316],{"class":11950,"line":12315},28,[42,12317,12038],{"emptyLinePlaceholder":12037},[42,12319,12321],{"class":11950,"line":12320},29,[42,12322,12324],{"class":12323},"sAwPA","\u002F\u002F Example usage\n",[42,12326,12328,12331,12334,12336,12339,12341,12343,12346],{"class":11950,"line":12327},30,[42,12329,12330],{"class":11954},"const",[42,12332,12333],{"class":11978}," lcg",[42,12335,12073],{"class":11954},[42,12337,12338],{"class":11954}," new",[42,12340,11959],{"class":11958},[42,12342,92],{"class":11962},[42,12344,12345],{"class":11978},"12345",[42,12347,12348],{"class":11962},");\n",[42,12350,12352,12355,12358,12361,12363],{"class":11950,"line":12351},31,[42,12353,12354],{"class":11962},"console.",[42,12356,12357],{"class":11958},"log",[42,12359,12360],{"class":11962},"(lcg.",[42,12362,12292],{"class":11958},[42,12364,12365],{"class":11962},"());\n",[42,12367,12369],{"class":11950,"line":12368},32,[42,12370,12371],{"class":12323},"\u002F\u002F Outputs a random integer\n",[42,12373,12375,12377,12379,12381,12384],{"class":11950,"line":12374},33,[42,12376,12354],{"class":11962},[42,12378,12357],{"class":11958},[42,12380,12360],{"class":11962},[42,12382,12383],{"class":11958},"nextFloat",[42,12385,12365],{"class":11962},[42,12387,12389],{"class":11950,"line":12388},34,[42,12390,12391],{"class":12323},"\u002F\u002F Outputs a random floating-point number between 0 and 1\n",[11934,12393,12395],{"id":12394},"code-to-test-randomness","Code to Test Randomness",[15,12397,12398],{},"We can test the randomness of the generator by generating a set of random numbers and plotting their distribution. Below is a simple test program:",[11939,12400,12402],{"className":11941,"code":12401,"language":11943,"meta":11944,"style":11944},"function testRandomness(\n  generator: LinearCongruentialGenerator,\n  iterations: number = 10000\n): void {\n  const results: number[] = new Array(10).fill(0);\n\n  for (let i = 0; i \u003C iterations; i++) {\n    const randomValue = Math.floor(generator.nextFloat() * 10);\n    results[randomValue]++;\n  }\n\n  console.log(\"Randomness Test Results:\");\n  results.forEach((count, index) => {\n    console.log(`${index}: ${\"*\".repeat(count \u002F 100)}`);\n  });\n}\n\n\u002F\u002F Example usage\nconst lcgTest = new LinearCongruentialGenerator(12345);\ntestRandomness(lcgTest);\n",[11946,12403,12404,12414,12425,12439,12450,12488,12492,12525,12555,12564,12568,12572,12587,12614,12653,12658,12662,12666,12670,12689],{"__ignoreMap":11944},[42,12405,12406,12409,12412],{"class":11950,"line":11951},[42,12407,12408],{"class":11954},"function",[42,12410,12411],{"class":11958}," testRandomness",[42,12413,12047],{"class":11962},[42,12415,12416,12419,12421,12423],{"class":11950,"line":11966},[42,12417,12418],{"class":11972},"  generator",[42,12420,10152],{"class":11954},[42,12422,11959],{"class":11958},[42,12424,12060],{"class":11962},[42,12426,12427,12430,12432,12434,12436],{"class":11950,"line":11985},[42,12428,12429],{"class":11972},"  iterations",[42,12431,10152],{"class":11954},[42,12433,11979],{"class":11978},[42,12435,12073],{"class":11954},[42,12437,12438],{"class":11978}," 10000\n",[42,12440,12441,12443,12445,12448],{"class":11950,"line":12002},[42,12442,112],{"class":11962},[42,12444,10152],{"class":11954},[42,12446,12447],{"class":11978}," void",[42,12449,11963],{"class":11962},[42,12451,12452,12455,12458,12460,12462,12465,12467,12469,12472,12474,12476,12479,12482,12484,12486],{"class":11950,"line":12018},[42,12453,12454],{"class":11954},"  const",[42,12456,12457],{"class":11978}," results",[42,12459,10152],{"class":11954},[42,12461,11979],{"class":11978},[42,12463,12464],{"class":11962},"[] ",[42,12466,88],{"class":11954},[42,12468,12338],{"class":11954},[42,12470,12471],{"class":11958}," Array",[42,12473,92],{"class":11962},[42,12475,4165],{"class":11978},[42,12477,12478],{"class":11962},").",[42,12480,12481],{"class":11958},"fill",[42,12483,92],{"class":11962},[42,12485,739],{"class":11978},[42,12487,12348],{"class":11962},[42,12489,12490],{"class":11950,"line":12034},[42,12491,12038],{"emptyLinePlaceholder":12037},[42,12493,12494,12497,12499,12502,12505,12507,12510,12513,12516,12519,12522],{"class":11950,"line":12041},[42,12495,12496],{"class":11954},"  for",[42,12498,12216],{"class":11962},[42,12500,12501],{"class":11954},"let",[42,12503,12504],{"class":11962}," i ",[42,12506,88],{"class":11954},[42,12508,12509],{"class":11978}," 0",[42,12511,12512],{"class":11962},"; i ",[42,12514,12515],{"class":11954},"\u003C",[42,12517,12518],{"class":11962}," iterations; i",[42,12520,12521],{"class":11954},"++",[42,12523,12524],{"class":11962},") {\n",[42,12526,12527,12530,12533,12535,12538,12541,12544,12546,12548,12550,12553],{"class":11950,"line":12050},[42,12528,12529],{"class":11954},"    const",[42,12531,12532],{"class":11978}," randomValue",[42,12534,12073],{"class":11954},[42,12536,12537],{"class":11962}," Math.",[42,12539,12540],{"class":11958},"floor",[42,12542,12543],{"class":11962},"(generator.",[42,12545,12383],{"class":11958},[42,12547,12295],{"class":11962},[42,12549,12224],{"class":11954},[42,12551,12552],{"class":11978}," 10",[42,12554,12348],{"class":11962},[42,12556,12557,12560,12562],{"class":11950,"line":12063},[42,12558,12559],{"class":11962},"    results[randomValue]",[42,12561,12521],{"class":11954},[42,12563,11982],{"class":11962},[42,12565,12566],{"class":11950,"line":12081},[42,12567,12181],{"class":11962},[42,12569,12570],{"class":11950,"line":12098},[42,12571,12038],{"emptyLinePlaceholder":12037},[42,12573,12574,12577,12579,12581,12585],{"class":11950,"line":12119},[42,12575,12576],{"class":11962},"  console.",[42,12578,12357],{"class":11958},[42,12580,92],{"class":11962},[42,12582,12584],{"class":12583},"sU2Wk","\"Randomness Test Results:\"",[42,12586,12348],{"class":11962},[42,12588,12589,12592,12595,12598,12601,12603,12606,12609,12612],{"class":11950,"line":12125},[42,12590,12591],{"class":11962},"  results.",[42,12593,12594],{"class":11958},"forEach",[42,12596,12597],{"class":11962},"((",[42,12599,12600],{"class":11972},"count",[42,12602,445],{"class":11962},[42,12604,12605],{"class":11972},"index",[42,12607,12608],{"class":11962},") ",[42,12610,12611],{"class":11954},"=>",[42,12613,11963],{"class":11962},[42,12615,12616,12619,12621,12623,12626,12628,12631,12633,12636,12638,12640,12643,12646,12648,12651],{"class":11950,"line":12139},[42,12617,12618],{"class":11962},"    console.",[42,12620,12357],{"class":11958},[42,12622,92],{"class":11962},[42,12624,12625],{"class":12583},"`${",[42,12627,12605],{"class":11962},[42,12629,12630],{"class":12583},"}: ${\"*\"",[42,12632,413],{"class":12583},[42,12634,12635],{"class":11958},"repeat",[42,12637,92],{"class":12583},[42,12639,12600],{"class":11962},[42,12641,12642],{"class":11954}," \u002F",[42,12644,12645],{"class":11978}," 100",[42,12647,112],{"class":12583},[42,12649,12650],{"class":12583},"}`",[42,12652,12348],{"class":11962},[42,12654,12655],{"class":11950,"line":12152},[42,12656,12657],{"class":11962},"  });\n",[42,12659,12660],{"class":11950,"line":12165},[42,12661,12312],{"class":11962},[42,12663,12664],{"class":11950,"line":12178},[42,12665,12038],{"emptyLinePlaceholder":12037},[42,12667,12668],{"class":11950,"line":12184},[42,12669,12324],{"class":12323},[42,12671,12672,12674,12677,12679,12681,12683,12685,12687],{"class":11950,"line":12189},[42,12673,12330],{"class":11954},[42,12675,12676],{"class":11978}," lcgTest",[42,12678,12073],{"class":11954},[42,12680,12338],{"class":11954},[42,12682,11959],{"class":11958},[42,12684,92],{"class":11962},[42,12686,12345],{"class":11978},[42,12688,12348],{"class":11962},[42,12690,12691,12694],{"class":11950,"line":12207},[42,12692,12693],{"class":11958},"testRandomness",[42,12695,12696],{"class":11962},"(lcgTest);\n",[15,12698,12699],{},"This test program generates a set of random numbers and counts the occurrences of each number. By observing the distribution of the output, we can make an initial assessment of the randomness of the generator.",[12701,12702,12703],"style",{},"html pre.shiki code .snl16, html code.shiki .snl16{--shiki-default:#F97583}html pre.shiki code .svObZ, html code.shiki .svObZ{--shiki-default:#B392F0}html pre.shiki code .s95oV, html code.shiki .s95oV{--shiki-default:#E1E4E8}html pre.shiki code .s9osk, html code.shiki .s9osk{--shiki-default:#FFAB70}html pre.shiki code .sDLfK, html code.shiki .sDLfK{--shiki-default:#79B8FF}html pre.shiki code .sAwPA, html code.shiki .sAwPA{--shiki-default:#6A737D}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html pre.shiki code .sU2Wk, html code.shiki .sU2Wk{--shiki-default:#9ECBFF}",{"title":11944,"searchDepth":11966,"depth":11966,"links":12705},[12706,12707,12708,12709,12710],{"id":12,"depth":11966,"text":13},{"id":1127,"depth":11966,"text":1128},{"id":4408,"depth":11966,"text":4409},{"id":11892,"depth":11966,"text":11893},{"id":11928,"depth":11966,"text":11929,"children":12711},[12712,12713],{"id":11936,"depth":11985,"text":11937},{"id":12394,"depth":11985,"text":12395},false,"md",null,{},"2024-09-11","\u002Fblog\u002F2024\u002F2024-09-11-linear-congruential-for-psedurandom-number",{"title":5,"description":11944},"blog\u002F2024\u002F2024-09-11-linear-congruential-for-psedurandom-number","Linear congruential method is a pseudo-random number generation method that generates a random number sequence based on a deterministic mathematical algorithm.",[12724,12725],"mathematics","cryptography","riIuMRcMixA_1ZLmyCvr4fXT7HyeupMuR2CTAHJQ-4A",1777090297052]