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 Python (3.4.3) (judge verdict: Time Limit Exceeded)

#from collections import deque
import numpy as np
import sys
sys.setrecursionlimit(10 ** 7)

read = sys.stdin.buffer.read
readline = sys.stdin.buffer.readline
readlines = sys.stdin.buffer.readlines

N, K = map(int, input().split())
A = np.array(input().split(), dtype=np.int64)


S = sum(A)
divisors = []

for i in range(1, S):
    if S < i * i:
        break

    if S % i == 0:
        divisors.append(i)
        divisors.append(S // i)


def check(x):
    modX = A % x
    modX = np.sort(modX)
    modX = modX[modX > 0]

    L = len(modX)
    rtn = False

    for i in range(L):
        cnt = 0
        minus = modX[:i]
        plus = modX[i:]

        cnt_minus = sum(minus - 0)
        cnt_plus = sum(x - plus)

        if cnt_minus <= K and cnt_plus <= K:
            rest = abs(cnt_minus - cnt_plus)
            if rest % x == 0:
                rtn = True
                break

    return rtn


ans = 1
for x in divisors:
    if check(x):
        ans = max(x, ans)

print(ans)