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: Accepted)

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];
    }
    
    for(int i = 1; i*i <= Asum; i++){
      if(Asum%i == 0 && i != Asum/i){
        x.add(i);
        x.add(Asum/i);
      }else if(Asum%i == 0){
        x.add(i);
      }
    }
    
    int len = x.size();
    
    
    int[] X = new int[len];
    
    for(int i = 0; i < len; i++){
      if(i*10 < (len*10)/2){
        X[i] = x.get(2*i);
      } else {
        X[i] = x.get(2*(len-i)-1);
      }
    }
    
    for(int i = len-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;
      
      for(int j = N-1; j > 0; j--){
        asub += a[j];
        Kcount += X[i]-a[j];
        if(asub+Kcount >= Asum-adsum*X[i]){
          break;
        }
      }
      
      if(Kcount <= K){
        result = X[i];
        break;
     }
    }

    System.out.println(result);
    
  }
}