UVALive - 6428（扩展欧几里德）

http://acm.hust.edu.cn/vjudge/problem/viewProblem.action?id=48388

前段时间偶然碰到的一道题，今天突然想到没把它记录下来。

比较不错的扩展欧几里德求解的应用

题意求是否满足ax+by=c gcd(a,b）==1 a>=0&&b>=0

数比较大 在求解时LL会溢出，用JAVA改写了一遍程序。

扩展欧几里德可以求出一组满足条件的解p= p1+b/gcd(a,b)*t q = q1+a/gcd(a,b)*t （t为任意整数）

因为上面的解有条件 所以根据p>=0 q>=0可以确定出来t的范围 在范围内枚举t可以求出解来 判断是否满足条件即可。

注意一下特殊情况，为0的情况。

 import java.text.*;
 import java.io.*;
 import java.util.*;
 import java.math.*;
 import java.applet.*;
 public class Main
 {
     static BigInteger x,y,o = BigInteger.valueOf(0),o1 = BigInteger.valueOf(1);
     static BigInteger gcd(BigInteger a,BigInteger  b)
     {
         if(b.compareTo(BigInteger.valueOf(0))==0)
             return a;
         else
             return gcd(b,a.mod(b));
     }
     static BigInteger exgcd(BigInteger a,BigInteger b)
     {
         if(b.compareTo(o)==0)
         {
             x = o1; y = o; return a;
         }
         BigInteger d = exgcd(b,a.mod(b));
         BigInteger temp = x;
         x = y;
         y = temp.subtract(a.divide(b).multiply(y))  ;
         return d;
     }
     public static void main(String[] args)
     {
         Scanner  cin = new Scanner(System.in);
         BigInteger a,b,c;
         while(cin.hasNext())
         {
             a = cin.nextBigInteger();
             b = cin.nextBigInteger();
             c = cin.nextBigInteger();
             if(c.compareTo(a)<0&&c.compareTo(b)<0)
             {
                 System.out.println("NO");
                 continue;
             }
             BigInteger o = BigInteger.valueOf(0);
             if(a.compareTo(o)==0||b.compareTo(o)==0)
             {
                 if(a.compareTo(o)==0&&b.compareTo(o)==0)
                 {
                     if(c.compareTo(o)==0)
                     System.out.println("YES");
                     else
                     System.out.println("NO");
                 }
                 else if(a.compareTo(o)==0)
                 {
                     if(c.mod(b).compareTo(o)==0)
                     System.out.println("YES");
                     else
                     System.out.println("NO");
                 }
                 else
                 {
                     if(c.mod(a).compareTo(o)==0)
                     System.out.println("YES");
                     else
                     System.out.println("NO");
                 }
                 continue;
             }
             if(c.compareTo(b)==0||c.compareTo(a)==0)
             {
                 System.out.println("YES");
                 continue;
             }
             BigInteger t = gcd(a,b);
             if(c.mod(t).compareTo(o)!=0)
             {
                 System.out.println("NO");
                 continue;
             }
             exgcd(a,b);
             x = x.multiply(c.divide(t));
             y = y.multiply(c.divide(t));
             BigInteger k1 = b.divide(t);
             BigInteger k2 = a.divide(t);
             int flag = 0;
             BigInteger tx=x,ty=y;
             BigInteger f1 = (x.multiply(BigInteger.valueOf(-1))).divide(k1).subtract(BigInteger.valueOf(1));
             BigInteger f2 = y.divide(k2).add(BigInteger.valueOf(1));
             BigInteger e = f1;
             while(e.compareTo(f2)<=0)
             {
                 tx = k1.multiply(e).add(x);
                 ty = k2.multiply(e);
                 ty = y.subtract(ty);
                 if(tx.compareTo(o1)>=0&&ty.compareTo(o1)>=0)
                 {
                     if(gcd(tx,ty).compareTo(BigInteger.valueOf(1))==0)
                     {
                         //System.out.println(tx+" "+ty);
                         flag = 1;
                         break;
                     }
                 }
                 e = e.add(BigInteger.valueOf(1));
             }
             if(flag==1)
             System.out.println("YES");
             else
             System.out.println("NO");
         }
     }
 }