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=list(map(int,input().split()))
sumA = sum(A)

divisors = []
for i in range(1, int(sumA**0.5)+1):
    if sumA % i ==0:
        divisors.append(i)
        if  i != sumA // i:
            divisors.append(sumA//i)

divisors.sort()

ans_temp = 0

for i in range(len(divisors)):
    listA = [0]*N
    for j in range(N):
        listA[j] = A[j] % divisors[i]
    listA.sort()
    #if divisors[i] > 4:
        #print(listA)
    for j in range(N):
        sumKp = 0
        sumKm = 0
        for k in range(len(listA)):
            if k >= j:
                sumKp += divisors[i] - listA[k]
            else:
                sumKm += listA[k]

        if sumKp > K:
            continue

        sumK = sumKp - sumKm
        #print(str(divisors[i]) + ':' +str(sumK) + ':' + str(sumKm))

        if sumK == 0 and divisors[i] > ans_temp:

            ans_temp = divisors[i]
            
    

print(ans_temp)