Euclidean algorithm and extension algorithm.

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Euclidean algorithm, also known as the greatest common divisor method, refers to the calculation of two positive integers, a, B. The fields of application are two aspects of mathematics and computer. The formula GCD (b) = gcd (b,a mod b).

Proof: R = a mod B, a = b * k + r;=> r = A-c * k;d|a && d|b=> (A/D-b/d * k) = R/D=>D|R∴GCD (A, B) = GCD (b, a mod b); code

int gcd (intint  b) {  return0 ? B:GCD (b, a% b);  }

The extended Euclidean algorithm is used to solve a set of X, y in known a, b to satisfy the Bézout equation: ax+by = gcd (A, B) =d (the solution must exist, according to the correlation theorem in number theory). Extended Euclidean is commonly used in solving linear equations and equations. Proof: a*x + b*y = gcd (A, B) b*x + (a mod b) *y = gcd (b, a mod b); gcd (A, b) = gcd (b, a mod b) =>a*x + b*y = b*x + (A-(A/b) *b *y) =>a*x + b*y = a*y + b (x-a/b *y) ∴ There is a set of solutions, x = y; y = x-a/b * y; the solution of XY is based on Abxy, and when a% b = = 0, you can get {x = 0, y = 1} This set of solutions, which is the recursive boundary. Using the computer to solve the code is

#include <iostream>using namespacestd;intgcdintAintb) {    return(a% b = =0? B:GCD (b, a%b));}voidFun (intAintBint&x,int&y) {    if(a% b = =0) {x=0; y =1; return; } fun (b, a%b, x, y); intx1 = x, y1 =y; X=Y1; Y= x1-a/b *Y1; return;}intMain () {intx, y; intA, B;  while(Cin >> a >>b) {Fun (A, B, X, y); cout<<"x ="<< x <<"y ="<< y <<Endl; }    return 0;}

Euclidean algorithm and extension algorithm.