The Balance POJ - 2142

首先，可以知道题目要求解一个\(ax+by=c\)的方程，且\(x+y\)最小。

感性证明:

当\(a>b\)时，\(y\)取最小正整数解，\(b\)减的多，\(a\)增的少，此时\(x+y\)取最小值。（类似比热容与温度之间）

反之亦然。

#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;
}