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.InputStreamReader;
import java.io.PrintStream;
import java.util.*;

public class Main {
    public static final Scanner in = new Scanner(new BufferedReader(new InputStreamReader(System.in)));
    public static final PrintStream out = System.out;
    public static final PrintStream err = System.err;

    public static void main(String[] args) {
        int N = in.nextInt();
        int K = in.nextInt();
        int[] A = new int[N];

        int sum = 0;
        for (int i = 0; i < N; i++) {
            sum += A[i] = in.nextInt();
        }

        err.println(sum);

        Set<Integer> P = new TreeSet<>();
        for (int i = 1; i * i <= sum; i ++) {
            if (sum % i == 0) {
                P.add(sum / i);
                P.add(i);
            }
        }

        int result = 1;
        for (int p : P) {
            int[] r = new int[N];
            int rs = 0;
            for (int i = 0; i < N; i++) {
                rs += r[i] = A[i] % p;
            }

            Arrays.sort(r);
            int ls = 0, s = rs;
            for (int i = N - 1; 0 <= i ; i--) {
                if (r[i] == 0) {
                    continue;
                }
                ls += p - r[i];
                s -= r[i];
                if (ls == s) {
                    break;
                }
            }

            err.printf("p: %d, r: %s, rs: %d, ls: %d, s: %d%n", p, Arrays.toString(r), rs, ls, s);

            if (s <= K) {
                result = Math.max(result, p);
            }
        }

        out.println(result);
    }
}