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: Wrong Answer)

import java.util.*;
public class Main {

	public static void main(String[] args) {
		Scanner sc = new Scanner(System.in);
		int N = sc.nextInt();
		int K = sc.nextInt();
		int[] A = new int[N];
		int sum = 0, ans = 0;
		for(int i = 0; i < N; i++) {
			A[i] = sc.nextInt();
			sum += A[i];
		}
		Set<Integer> cd = new HashSet<>();
		for(int i = 1; i <= Math.sqrt(sum); i++)
			if(sum % i == 0) {
				cd.add(i);
				cd.add(sum/i);
			}
		for(int i : cd) {
			int min = Integer.MAX_VALUE;
			int[] B = new int[N];
			for(int j = 0; j < N; j++)
				B[j] = A[j] % i;
			Arrays.sort(B);
			int[] M = new int[N+1];
			int[] P = new int[N+1];
			for(int j = 0; j < N; j++)
				M[j+1] = M[j] + B[j];
			for(int j = N-1; j >= 0; j--)
				if(B[j] != 0)
					P[j] = P[j+1] + (i - B[j]);
			for(int j = 1; j < N; j++)
				if((M[j] - P[j]) % i == 0)
					min = Math.min(min, Math.max(M[j], P[j]));
			if(min <= K)
				ans = Math.max(ans, i);
		}
		System.out.println(ans);
	}

}