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 PyPy3 (2.4.0) (judge verdict: Runtime Error)

INF = float("inf")
n, k = map(int, input().split())
a = list(map(int, input().split()))

def get_divisors(num):
    f_divs, l_divs = [], []
    for i in range(1, int(num**0.5)+1):
        if not num % i:
            f_divs.append(i)
            if i != num // i:
                l_divs.append(num // i)
    return f_divs + l_divs[::-1]

s = sum(a)
divs = get_divisors(s)
ans = 1

def evl(d):
    if d == 1: return True
    m = d*n*2
    rem = [x % d for x in a]
    dp = [[INF] * (m+d*2+10) for _ in range(n+1)]
    mid = m // 2
    dp[0][mid] = 0
    for i in range(n):
        for j in range(m):
            if rem[i] == 0:
                dp[i+1][j] = min(dp[i+1][j], dp[i][j])
                continue
            dp[i+1][j-rem[i]] = min(dp[i+1][j-rem[i]], dp[i][j]+rem[i])
            dp[i+1][j+d-rem[i]] = min(dp[i+1][j+d-rem[i]], dp[i][j]+(d-rem[i]))
    return dp[n][mid] // 2 <= k

l, r = -1, len(divs)

while r-l > 1:
    x = (r+l) // 2
    if evl(divs[x]): l = x
    else: r = x

print(divs[l])