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)

# -*- coding: utf-8 -*-

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

sum_a = sum(A)
# dc:Divisor Candidate(割る数の候補)
dc_list = [x for x in reversed(range(1, sum_a + 1)) if sum_a % x == 0]

for d in dc_list:
    mods = sorted([a % d for a in A])
    # cs:sumulative sum(累積和)
    mod_cs = [0] * (len(mods) + 1)
    for i, mod in enumerate(mods):
        mod_cs[i + 1] = mod_cs[i] + mod

    for i, mod_cumulative in enumerate(mod_cs):
        minus_op = mod_cumulative
        plus_op = (d * (len(mod_cs) - 1 - i)) - (mod_cs[-1] - mod_cs[i])

        if minus_op == plus_op and minus_op <= K:
            print(d)
            exit(0)
        elif minus_op > K:
            break