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;
}