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: Wrong Answer)

import sys

def solve():
    input = sys.stdin.readline
    N, K = map(int, input().split())
    A = [int(a) for a in input().split()]
    for i in reversed(range(1, 1 + 10 ** 6)):
        if i % 2 == 1:
            plus, minus = 0, 0
            for a in A:
                m = a % i
                if m > i // 2: plus += (i - m)
                elif m > 0: minus += m
            if abs(plus - minus) % i == 0:
                if plus + minus + abs(plus - minus) // i <= K:
                    print(i)
                    break
        else:
            plus, minus, both = 0, 0, 0
            for a in A:
                m = a % i
                if m > i // 2: plus += (i - m)
                elif 0 < m < i // 2: minus += m
                elif m == i // 2: both += 1
            if abs(plus - minus) % i == 0:
                if both % 2 == 0 and plus + minus + abs(plus - minus) // i + (both // 2) * (i // 2) <= K:
                    print(i)
                    break
            elif abs(plus - minus) % i == i // 2 and plus + minus + abs(plus - minus) // i + (both // 2) * (i // 2) + (i // 2) <= K:
                if both % 2 == 1:
                    print(i)
                    break
    else: print(1)
    return 0

if __name__ == "__main__":
    solve()