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。

  1 import java.text.*;

  2 import java.io.*;

  3 import java.util.*;

  4 import java.math.*;

  5 import java.applet.*;

  6 public class Main 

  7 {

  8     static BigInteger x,y,o = BigInteger.valueOf(0),o1 = BigInteger.valueOf(1);

  9     static BigInteger gcd(BigInteger a,BigInteger  b)

 10     {

 11         if(b.compareTo(BigInteger.valueOf(0))==0)

 12             return a;

 13         else

 14             return gcd(b,a.mod(b));

 15     }

 16     static BigInteger exgcd(BigInteger a,BigInteger b)

 17     {

 18         if(b.compareTo(o)==0)

 19         {

 20             x = o1; y = o; return a;

 21         }

 22         BigInteger d = exgcd(b,a.mod(b));

 23         BigInteger temp = x;

 24         x = y;

 25         y = temp.subtract(a.divide(b).multiply(y))  ;

 26         return d;

 27     }

 28     public static void main(String[] args)

 29     {

 30         Scanner  cin = new Scanner(System.in);

 31         BigInteger a,b,c;

 32         while(cin.hasNext())

 33         {

 34             a = cin.nextBigInteger();

 35             b = cin.nextBigInteger();

 36             c = cin.nextBigInteger();

 37             if(c.compareTo(a)<0&&c.compareTo(b)<0)

 38             {

 39                 System.out.println("NO");

 40                 continue;

 41             }

 42             BigInteger o = BigInteger.valueOf(0);

 43             if(a.compareTo(o)==0||b.compareTo(o)==0)

 44             {

 45                 if(a.compareTo(o)==0&&b.compareTo(o)==0)

 46                 {

 47                     if(c.compareTo(o)==0)

 48                     System.out.println("YES");

 49                     else

 50                     System.out.println("NO");

 51                 }

 52                 else if(a.compareTo(o)==0)

 53                 {

 54                     if(c.mod(b).compareTo(o)==0)

 55                     System.out.println("YES");

 56                     else

 57                     System.out.println("NO");

 58                 }

 59                 else

 60                 {

 61                     if(c.mod(a).compareTo(o)==0)

 62                     System.out.println("YES");

 63                     else

 64                     System.out.println("NO");

 65                 }

 66                 continue;

 67             }

 68             if(c.compareTo(b)==0||c.compareTo(a)==0)

 69             {

 70                 System.out.println("YES");

 71                 continue;

 72             }

 73             BigInteger t = gcd(a,b);

 74             if(c.mod(t).compareTo(o)!=0)

 75             {

 76                 System.out.println("NO");

 77                 continue;

 78             }

 79             exgcd(a,b);

 80             x = x.multiply(c.divide(t));

 81             y = y.multiply(c.divide(t));

 82             BigInteger k1 = b.divide(t);

 83             BigInteger k2 = a.divide(t);

 84             int flag = 0;

 85             BigInteger tx=x,ty=y;

 86             BigInteger f1 = (x.multiply(BigInteger.valueOf(-1))).divide(k1).subtract(BigInteger.valueOf(1));

 87             BigInteger f2 = y.divide(k2).add(BigInteger.valueOf(1));

 88             BigInteger e = f1;

 89             while(e.compareTo(f2)<=0)

 90             {

 91                 tx = k1.multiply(e).add(x);

 92                 ty = k2.multiply(e);

 93                 ty = y.subtract(ty);

 94                 if(tx.compareTo(o1)>=0&&ty.compareTo(o1)>=0)

 95                 {

 96                     if(gcd(tx,ty).compareTo(BigInteger.valueOf(1))==0)

 97                     {

 98                         //System.out.println(tx+" "+ty);

 99                         flag = 1;

100                         break;

101                     }

102                 }

103                 e = e.add(BigInteger.valueOf(1));

104             }

105             if(flag==1)

106             System.out.println("YES");

107             else

108             System.out.println("NO");

109         }

110     }

111 }

View Code