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 C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <bits/stdc++.h>
using namespace std;

int main() {
    ios::sync_with_stdio(0); cin.tie(0);
    int n, k;
    cin >> n >> k;
    vector<int> a(n);
    int sum = 0;
    for(int i = 0; i < n; ++i) {
        cin >> a[i];
        sum += a[i];
    }

    vector<int> div;
    for(int i = 1; i * i <= sum; ++i) {
        if(sum % i == 0) {
            int j = sum / i;
            div.push_back(i);
            div.push_back(j);
        }
    }
    sort(div.rbegin(), div.rend());
    for(int d : div) {
        vector<int> v(n);
        int t = 0;
        for(int i = 0; i < n; ++i) {
            v[i] = a[i] % d;
            t += d - v[i];
        }
        t /= d;
        sort(begin(v), end(v));
        int mov = 0;
        for(int i = 0; i < t; ++i) {
            mov += v[i];
        }
        if(mov <= k) {
            cout << d << '\n';
            return 0;
        }
    }
    assert(0 == 1);
}