Extended Euclidean algorithm-Extended Euclidean Algorithm

The euclidean algorithm, also known as the moving phase division, is used to calculate the maximum approximate number of two integers A and B. The computation principle depends on the following theorem: The GCD function is used to calculate the maximum approximate number of (a, B. Basic properties of the gcd function: gcd (a, B) = gcd (B, A) = gcd (-a, B) = gcd (| A |, | B |) the formula expression gcd (a, B) = gcd (B, A mod B) proves that a can be expressed as a = kb + R, then R = a mod B Assumes that D is, d | a, d | B, r = A-kb, so d | r is (B, A mod B) assume that D is (B, A mod B), d | B, d | r, but a = kb + R, so d is (a, B) so (A, B) and (B, A mod B) are the same, and the maximum number of them must be equal.

Extended:

Expressed in mathematical language: Calculate the integer x and y to make AX + by = C. it can be found that if gcd (a, B) | C is not true, that is, C is not a multiple of the maximum common divisor of A and B, there is obviously no integer solution.

Therefore, we generally calculate AX + by = gcd (a, B), as long as we obtain an integer solution (x, y), then any C (except gcd (A, B )) you can get it by multiplying the two sides by a multiple.

Code:

#include <iostream>using namespace std;int ext_gcd(int a,int b,int &x,int &y){if(b==0){x=1;y=0;return a;}int d=ext_gcd(b,a%b,y,x);y=y-(a/b)*x;//  x=_y;return d;//  y=_x-(a/b)*_y;} int main(){int x,y,a=99,b=78;cout<<ext_gcd(a,b,x,y)<<endl;cout<<x<<" "<<y<<endl;}

Note:

Assume that we have obtained:

_ D = gcd (B, A mod B) and _ d = B * _ x + (a mod B) * _ y (_ d, _ x, _ y) for recursive procedures, there are

D = gcd (a, B) = _ d = gcd (B, A mod B), in order to obtain the X and Y that meet d = AX +, use the preceding equation d = _ d for Deformation:

D = B * _ x + (a-(A/B) * _ y = a * _ y + B (_ x-(A/B) * _ y)

When x = _ y and Y = _ x-(A/B) * _ y are selected, the equation can be satisfied. In this way, recursion can proceed. Of course, the conditions for recursion termination are: B = 0, return ()