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)

N, K = map(int, input().split())
A = list(map(int, input().split()))

def divisors(n):
    ds = []
    for d in range(1, n + 1):
        if d * d > n:
            break
        if n % d == 0:
            ds.append(d)
            if d * d < n:
                ds.append(n // d)
    return ds

sum_A = sum(A)
ds = sorted(divisors(sum_A), reverse=True)

for d in ds:
    pos = []
    neg = []
    n_both = 0
    cost = 0
    for a in A:
        r = a % d
        if r < d - r:
            neg.append(r)
        elif r > d - r:
            pos.append(d - r)
        else:
            n_both += 1
    if n_both >= 2:
        cost += (n_both // 2) * (d // 2)
        n_both %= 2
    sum_pos = sum(pos)
    sum_neg = sum(neg)
    if sum_pos < sum_neg:
        cost += sum_pos
        sum_neg -= sum_pos
        if n_both:
            cost += sum_neg
        else:
            cost += sum_neg - max(neg)
    elif sum_pos > sum_neg:
        cost += sum_neg
        sum_pos -= sum_neg
        if n_both:
            cost += sum_pos
        else:
            cost += sum_pos - max(pos)
    else:
        assert n_both == 0
        cost += sum_pos
    if cost <= K:
        print(d)
        break