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(c) for c in input().split()]
S = sum(A)
pls = []
for i in range(1,S):
  if i*i>S:
    break
  if S%i==0:
    pls += [i]
    pls += [S//i]
pls.sort(reverse=True)
for p in pls:
    B = []
    for i in range(N):
      B += [A[i]%p]
    B.sort()
    for j in range(N):
      if B[j]!=0:
        break
    m = j
    lim = N-1
    cnt = 0
    while m<lim:
      if B[m]<p-B[-1]:
        B[-1] += B[m]
        cnt += B[m]
        B[m] = 0
        m += 1
      elif B[m]>p-B[-1]:
        cnt += p-B[-1]
        B.pop()
        B[m] -= p-B[-1]
        lim -= 1
      else:
        B.pop()
        cnt += B[m]
        B[m] = 0
        m += 1
        lim -= 1
    if B[-1]==0 and cnt<=K:
      print(p)
      break