The Balance 扩展欧几里得

题意：给定a,b,c;求a*x+b*y=c；

思路：典型的 扩展欧几里得，主要是如何求最小和；

a*x+b*y=c,特解x0，y0。

通解 x=x0+b/gcd*t ; y=y0-a/gcd*t  ;|x|+|y|=|x0+b/gcd*t |+|y0-a/gcd*t |，

令a>b(如果不是，交换)，因为y的减少大于x的增长，所以当t=y0/(a/gcd)时，和最小。

如果不能整除，也是t左右两个数。

#include<cstdio>#include<cstring>#include<algorithm>#include<cmath>using namespace std;typedef long long ll;int egcd(int a,int b,int &x,int &y){    if(b==0){        x=1;y=0;        return a;    }    int d=egcd(b,a%b,y,x);    y-=(a/b)*x;    return d;}int main(){    int a,b,c,d,x,y,t;    int x1,x2,y1,y2;    while(scanf("%d%d%d",&a,&b,&c)&&a||b||c)    {        bool flag=true;        if(a<b) {flag=false;swap(a,b);}        d=egcd(a,b,x,y);        a/=d;b/=d;c/=d;        x=c*x;y=c*y;        t=y/a;        while(y-a*t<0)//y<0时，y-a*t<0，t应该-1，让y-a*t>0,不然t增加绝对值只会越来越大        t--;        x1=abs(x+b*t);y1=abs(y-a*t);        t++;        x2=abs(x+b*t);y2=abs(y-a*t);        if((x1+y1<x2+y2)||((x1+y1==x2+y2)&&(x1*a+y1*b<x2*a+y2*b)))        {            x=x1;y=y1;        }        else {x=x2;y=y2;}        if(flag)  printf("%d %d\n",x,y);        else  printf("%d %d\n",y,x);    }}

Ms. Iyo Kiffa-Australis has a balance and only two kinds of weights to measure a dose of medicine. For example, to measure 200mg of aspirin using 300mg weights and 700mg weights, she can put one 700mg weight on the side of the medicine and three 300mg weights on the opposite side (Figure 1). Although she could put four 300mg weights on the medicine side and two 700mg weights on the other (Figure 2), she would not choose this solution because it is less convenient to use more weights. 
You are asked to help her by calculating how many weights are required. 

Input

The input is a sequence of datasets. A dataset is a line containing three positive integers a, b, and d separated by a space. The following relations hold: a != b, a <= 10000, b <= 10000, and d <= 50000. You may assume that it is possible to measure d mg using a combination of a mg and b mg weights. In other words, you need not consider "no solution" cases. 
The end of the input is indicated by a line containing three zeros separated by a space. It is not a dataset.

Output

The output should be composed of lines, each corresponding to an input dataset (a, b, d). An output line should contain two nonnegative integers x and y separated by a space. They should satisfy the following three conditions. 

• You can measure dmg using x many amg weights and y many bmg weights. 
  
• The total number of weights (x + y) is the smallest among those pairs of nonnegative integers satisfying the previous condition. 
  
• The total mass of weights (ax + by) is the smallest among those pairs of nonnegative integers satisfying the previous two conditions.

No extra characters (e.g. extra spaces) should appear in the output.

Sample Input

700 300 200500 200 300500 200 500275 110 330275 110 385648 375 40023 1 100000 0 0

Sample Output

1 31 11 00 31 149 743333 1