8.14 Number Theory - Extended Euclid's theorem and congruence equation

Extended Euclid's theorem for the solution of the equation ax + by = gcd (a, b)

int Exgcd(int a, int b, int &x, int &y) {
  if (!b) {
    x = 1;
    y = 0;
    return a;
  }
  int d = Exgcd(b, a % b, x, y);
  int t = x;
  x = y;
  y = t - (a / b) * y;
  return d;
}

See proof: https://oi-wiki.org/math/gcd/

By constructing the equivalent equation Euclid's theorem, a recursive solution;

 

Linear congruent equation:

Shaped like ax ≡ c (mod b) of the equation;

1. equivalent: equivalent to ax + by = c, on both sides at the same time as the modulo

2. determination condition: ax by Solution + = gcd (a, b) of exgcd can be evaluated, so that c is gcd (a, b) is an integer multiple of, when AX + by = c solvable equation

3. General Solution: After solving the general solution can be configured according to the original equation: x + k * b / gcd, y - k * a / gcd

4. The smallest positive integer solutions: The general solution in the form of solutions available smallest positive integer x is (x% t + t)% t, t = b / gcd;

int ex_gcd(int a, int b, int& x, int& y) {
  if (b == 0) {
    x = 1;
    y = 0;
    return a;
  }
  int d = ex_gcd(b, a % b, x, y);
  int temp = x;
  x = y;
  y = temp - a / b * y;
  return d;
}
bool liEu(int a, int b, int c, int& x, int& y) {
  int d = ex_gcd(a, b, x, y);
  if (c % d != 0) return 0;
  int k = c / d;
  x *= k;
  y *= k;
  return 1;
}