HDOJ 1098 Ignatius's puzzle

Problem Description

Ignatius is poor at math,he falls across a puzzle problem,so he has no choice but to appeal to Eddy. this problem describes that:f(x)=5*x^13+13*x^5+k*a*x,input a nonegative integer k(k<10000),to find the minimal nonegative integer a,make the arbitrary integer x ,65|f(x)if

no exists that a,then print “no”.

Input

The input contains several test cases. Each test case consists of a nonegative integer k, More details in the Sample Input.

Output

The output contains a string “no”,if you can’t find a,or you should output a line contains the a.More details in the Sample Output.

Sample Input

11

100

9999

Sample Output

22

no

43

题目大意：方程f(x)=5*x^13+13*x^5+k*a*x；输入任意一个数k，是否存在一个数a，对任意x都能使得f(x)能被65整除。

现假设存在这个数a ,因为对于任意x方程都成立

所以，当x=1时f(x)=18+ka

又因为f(x)能被65整出，故设n为整数

可得，f(x)=n*65;

即：18+ka=n*65;

因为n为整数，若要方程成立

则问题转化为，

对于给定范围的a只需要验证，

是否存在一个a使得(18+k*a)%65==0

所以容易解得

注意，这里有童鞋不理解为什么a只需到65即可

因为，当a==66时

也就相当于已经找了一个周期了，所以再找下去也找不到适当的a了

import java.util.Scanner;
public class Main {
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        while(sc.hasNext()){
            int k= sc.nextInt();
            boolean flag=false;
            for(int a=0;a<=65;a++){
                if((18+k*a)%65==0){
                    System.out.println(a);
                    flag = true;
                    break;
                }
            }
            if(!flag){
                System.out.println("no");
            }
        }
    }
}