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 PyPy2 (5.6.0) (judge verdict: Wrong Answer)

import sys
sys.setrecursionlimit(10**7)

def factorize(n):
    d = 2
    ans = []
    while d * d <= n:
        e = 0
        while n % d == 0:
            n /= d
            e += 1
        if e:
            ans.append((d,e))
        d += 1 + d%2
    if n > 1:
        ans.append((n, 1))
    return ans

from itertools import product
def divisors(N):
    facs = factorize(N)
    bases, expos = zip(*facs)
    for es in product(*[range(e+1) for e in expos]):
        ans = 1
        for i, e in enumerate(es):
            ans *= bases[i] ** e
        yield ans

def memo(f):
    class memodict(dict):
        def __init__(self, f):
            self.f = f
        def __call__(self, *args):
            return self[args]
        def __missing__(self, key):
            ret = self[key] = self.f(*key)
            return ret
    return memodict(f)

def solve(N, K, A):
    S = sum(A)
    def check(g):
        items = []
        for x in A:
            positive = (-x)%g
            negative = g + x if x < g else x % g
            items.append((positive, negative))
        # does there exist a coloring so that sumpos and sumneg <= K
        @memo
        def dp(i, sumpos, sumneg):
            if sumpos > K or sumneg > K: return False
            if i == len(items): return True
            p, n = items[i]
            return dp(i+1, sumpos + p, sumneg) or dp(i+1, sumpos, sumneg + n)
        return dp(0,0,0)
    
    for g in sorted(divisors(S), reverse = True):
        if check(g): return g
    return 1
        


N, K = map(int, raw_input().split())
A = map(int, raw_input().split())
print solve(N, K, A)