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)

N, K = map(int, input().split())
A = [int(i) for i in input().split()]
A.sort()

s = sum(A)
i = 1
q1 = []
q2 = []
while i * i <= s :
    if s % i == 0 :
        q1.append(i)
        if i * i != s :
            q2.append(s//i)
    i += 1
prime = q2 + q1[::-1]

def solve() :
    for p in prime :
        m = [a%p for a in A]
        m.sort()
        
        dpl = [0] * N
        dpr = [0] * N
        
        dpl[N-1] = p - m[N-1]
        dpr[0] = m[0]
        for i in range(N-2, -1, -1) :
            dpl[i] = dpl[i+1] + p - m[i]
        for i in range(1, N) :
            dpr[i] = dpr[i-1] + m[i]
        
        for i in range(1, N) :
            if dpl[i] == dpr[i-1] and dpl[i] <= K :
                return p
            
print(solve())