Problem:
Score : 500 points

Problem Statement

We have a sequence of N integers: A_1, A_2, \cdots, A_N.

You can perform the following operation between 0 and K times (inclusive):

Choose two integers i and j such that i \neq j, each between 1 and N (inclusive). Add 1 to A_i and -1 to A_j, possibly producing a negative element.

Compute the maximum possible positive integer that divides every element of A after the operations. Here a positive integer x divides an integer y if and only if there exists an integer z such that y = xz.

Constraints

2 \leq N \leq 500

1 \leq A_i \leq 10^6

0 \leq K \leq 10^9

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N K
A_1 A_2 \cdots A_{N-1} A_{N}

Output

Print the maximum possible positive integer that divides every element of A after the operations.

Sample Input 1

2 3
8 20

Sample Output 1

7

7 will divide every element of A if, for example, we perform the following operation:

Choose i = 2, j = 1. A becomes (7, 21).

We cannot reach the situation where 8 or greater integer divides every element of A.

Sample Input 2

2 10
3 5

Sample Output 2

8

Consider performing the following five operations:

Choose i = 2, j = 1. A becomes (2, 6).

Choose i = 2, j = 1. A becomes (1, 7).

Choose i = 2, j = 1. A becomes (0, 8).

Choose i = 2, j = 1. A becomes (-1, 9).

Choose i = 1, j = 2. A becomes (0, 8).

Then, 0 = 8 \times 0 and 8 = 8 \times 1, so 8 divides every element of A. We cannot reach the situation where 9 or greater integer divides every element of A.

Sample Input 3

4 5
10 1 2 22

Sample Output 3

7

Sample Input 4

8 7
1 7 5 6 8 2 6 5

Sample Output 4

5

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Time Limit Exceeded)

import java.util.ArrayList;
import java.util.Arrays;
import java.util.List;
import java.util.*;
 
public class Main {
  public static void main (String[] arg) {
    Scanner scan = new Scanner(System.in);
    int N = scan.nextInt();
    int K = scan.nextInt();
    int A[] = new int[N];
    int Asum = 0;
    int a[] = new int[N];
    int ad[] = new int[N];
    List<Integer> x = new ArrayList<>();
    int result = 0;
    
    
    for(int i = 0; i < N; i++){
      A[i] = scan.nextInt();
      Asum += A[i];
    }
    x.add(Asum);
    for(int i = Asum/2; i >= 1; i--){
      if(Asum%i == 0){
        x.add(i);
      }
    }
    
    int[] X = new int[x.size()];
    for(int i = 0; i < x.size(); i++){
      X[i] = x.get(i);
    }
    
    Arrays.sort(X);
    
    for(int i = x.size()-1; i >= 0; i--){
      
    int asum = 0;
    int adsum = 0;
      
      for(int j = 0; j < N; j++){
        a[j] = A[j]%X[i];
        ad[j] = A[j]/X[i];
        asum += a[j];
        adsum += ad[j];
      }
      
      Arrays.sort(a);
      
      int asub = 0;
      int Kcount = 0;
      
      Ask_K: for(int j = N-1; j > 0; j--){
        asub += a[j];
        while(a[j] <= X[i]){

          if(asub+Kcount >= Asum-adsum*X[i]){
            break Ask_K;
          }
                    a[j]++;
          Kcount++;
        }
      }
      if(Kcount <= K){
        result = X[i];
        break;
     }
    }
    
    System.out.println(result);
    
  }
}