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: Wrong Answer)

def main():
    N, K = map(int, input().split())
    A = list(map(int, input().split()))
    m = sum(A)
    d = []
    for i in range(1, int(m**0.5)+1):
        if m % i == 0:
            d.append(i)
            if i != m // i:
                d.append(m//i)
    d.sort(reverse=True)
    for j in d:
        if j > K + max(A):
            continue
        p = 0
        m = 0
        d = 0
        for i in A:
            a =  i % j
            b = j - (i % j)
            if a == b:
                d += a
            elif a < b:
                p += a
            else:
                m += b
        if ((p + d) - m) % j == 0 and p + d <= 2 * K and m <= 2 * K:
            print(j)
            return
        if ((m + d) - p) % j == 0 and m + d <= 2 * K and p <= 2 * K:
            print(j)
            return
    print(1)
main()