First, it is possible to know a subject to be solved \ (ax + by = c \ ) equation, and \ (x + y \) minimum.
Inductive Proof:
When \ (a> b \) when, \ (Y \) takes the smallest positive integer solution, \ (B \) Save multiple, \ (A \) increased less, at this time \ (x + y \ ) takes a minimum value. (Similar to the ratio between temperature and heat capacity)
and vice versa.
#include<iostream>
#include<cmath>
#include<cstdlib>
using namespace std;
int a, b, c;
int exgcd(int a, int b, int &x, int &y) {
if(b == 0) {
x = 1, y = 0;
return a;
}
int g = exgcd(b, a % b, x, y);
int tmp = x;
x = y;
y = tmp - (a / b) * y;
return g;
}
void solve(int &x, int &y, int a, int b, int c) {
int g = exgcd(a, b, x, y);
x *= (c / g);
int t = b / g;
x = (x % t + t) % t;
y = (a * x - c) / b;
y = abs(y);
}
int main() {
while(cin >> a >> b >> c) {
if(!a && !b && !c) break;
int x1, y1, x2, y2;
solve(x1, y1, a, b, c);
solve(x2, y2, b, a, c);
if(x1 + y1 < x2 + y2) cout << x1 << " " << y1 << "\n";
else cout << y2 << " " << x2 << "\n";
}
return 0;
}