Extended Euclidean algorithm and its implementation

Euclidean algorithm, namely the greatest common divisor method, is used to find the integer a, b.

Euclidean algorithm C + + implementation code: (no need to determine a, b size relationship)

Long Long gcd (long long A,long long b) {return B?GCD (b,a%b): A;}

Extended Euclidean algorithm: Set A and B are not all 0, there is an integer x and y, making gcd (A, b) = xa + YB

Proof: Suppose A>b

When b==0: gcd (A, b) = A, at this time x=1, y=0

When ab!=0:

Set: x1a + y1b = gcd (A, B)

x2b + y2 (a%b) = gcd (b,a%b)

Because gcd (A, b) = = GCD (b,a%b)

So x1a + y1b = x2b + y2 (a%b) = x2b + y2 (a-a/b*b) =y2a + x2b-y2 (A/b) *b

So x1 = y2, y1 = x2-(A/b) y2

This can be based on x2, y2 to find out the solution of X1, y1.

Extended Euclid C + + implementation code 01:

Long Long exgcd (long long A,long long b,long long& x,long long& y) {    if (b==0) {        x=1;        y=0;        return A;    }    Long Long D=EXGCD (b,a%b,x,y);    Long long tmp=x;    x=y;    Y=tmp-a/b*y;    return D;}

Extended Euclid C + + implementation code 02:

#define LL Long longvoid exgcd (ll a,ll b,ll& d,ll& x,ll& y) {    if (!b) {D=a;x=1;y=0;return;}    EXGCD (b,a%b,d,y,x);    Y-= a/b*x;    return;}

Extended Euclidean algorithm and its implementation