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.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;
import java.io.PrintWriter;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.List;
import java.util.regex.Pattern;

public class Main {
    public void solve(BufferedReader stdin, PrintWriter stdout) throws NumberFormatException, IOException {
        Pattern space = Pattern.compile(" ");
        int[] line = space.splitAsStream(stdin.readLine()).mapToInt(Integer::parseInt).toArray();
        int n = line[0];
        int k = line[1];
        long[] a = space.splitAsStream(stdin.readLine()).mapToLong(Long::parseLong).toArray();

        long sum = Arrays.stream(a).sum();
        List<Long> factors = new ArrayList<>();
        for (long x = 1; x * x <= sum; x++) {
            if (sum % x == 0) {
                factors.add(x);
                factors.add(sum / x);
            }
        }

        long ans = 1;
        for (long d : factors) {
            long[] c = new long[n];
            for (int x = 0; x < n; x++) {
                c[x] = a[x] % d;
            }
            Arrays.sort(c);

            long[] b1 = new long[n + 1];
            long[] b2 = new long[n + 1];
            for (int i = 0; i < n; i++) {
                b1[i + 1] = b1[i] + c[i];
                b2[i + 1] = b2[i] + (d - c[i]);
            }

            for (int x = 0; x <= n; x++) {
                long p = b1[x];
                long q = b2[n] - b2[x];
                if (p == q && p <= k) {
                    ans = Math.max(ans, d);
                }
            }
        }
        stdout.println(ans);
    }

    public static void main(String[] args) throws NumberFormatException, IOException {
        BufferedReader stdin = new BufferedReader(new InputStreamReader(System.in));
        PrintWriter stdout = new PrintWriter(System.out, false);
        new Main().solve(stdin, stdout);
        stdout.flush();
    }
}