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 (Clang 3.8.0) (judge verdict: Accepted)

#include <iostream>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <algorithm>
#include <string>
#include <sstream>
#include <complex>
#include <vector>
#include <list>
#include <queue>
#include <deque>
#include <stack>
#include <map>
#include <set>
#include <climits>
#include <random>
#include <iomanip>

using namespace std;
typedef long long unsigned int ll;
typedef pair<int, int> P;

#define EPS (1e-7)
#define INF (1e9)
#define PI (acos(-1))

#define fillInt(xs, x)        \
  for (int i = 0; i < (x); i++) \
    scanf("%d", &xs[i]);
#define fillLong(xs, x)       \
  for (int i = 0; i < (x); i++) \
    scanf("%ld", &xs[i]);
#define fillString(xs, x)       \
  for (int i = 0; i < (x); i++) \
    cin >> xs[i];
#define sortv(xs) sort(xs.begin(), xs.end())
#define sortvdi(xs) sort(xs.begin(), xs.end(), std::greater<long>())
#define lbv(xs, x) lower_bound(xs.begin(), xs.end(), x) - xs.begin()
#define ubv(xs, x) upper_bound(xs.begin(), xs.end(), x) - xs.begin()
#define bs(xs, x) binary_search(xs.begin(), xs.end(), x)

#define rep(i,n) for(auto i=0; i<(n); i++)
#define isValidPoint(x, y, mx, my) x >= 0 && x  < mx && y >= 0 && y < my

int dx[4] = {1, 0, -1, 0}, dy[4] = {0, 1, 0, -1};

const int MOD = 1000000007;

struct Pony
{
    long a, b;
 
    Pony(long a_, long b_)
    {
        a = a_;
        b = b_;
    }
 
    bool operator<(const Pony &another) const
    {
        return b < another.b;
    };
};

int fac(int k){
    int sum = 0;
    for (int i = 1; i <= k; ++i)
    {
        sum += i;
    }
    return sum;
}

int main()
{
  cin.tie(0);
  ios::sync_with_stdio(false);

  long n, k;

  cin >> n >> k;

  long sm = 0;
  vector<long> v(n);
  fillLong(v, n);

  rep(i, n) {
    sm += v[i];
  }

  long res = 1;

  vector<long> yakusu;

  for(long i = 1; i <= sqrt(sm); i++) {
    if(!(sm % i)) {
      yakusu.push_back(i);
      yakusu.push_back(sm/i);
    }
  }

  sortv(yakusu);

  rep(i, yakusu.size()) {
    long y = yakusu[i];
    long cnt = 0;
    vector<long> v2(n);
    rep(j, n) {
      v2[j] = (v[j] + y) % y;
    }
    sortv(v2);
    long left = 0;
    long right = n-1;
    while(left < right) {
      long dff = v2[left];
      cnt += dff;
      left++;
      while(dff) {
        long tmp = y-v2[right];
        v2[right] = min(y, v2[right] + dff);
        dff -= min(dff, tmp);
        if (v2[right] == y) right--;
      }
    }
    if (cnt <= k) res = y;
  }
  
  cout << res << endl;
}