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: Runtime Error)

def yakusu(n):
    m=1
    re=[]
    while m**2<n:
        a,md=divmod(n,m)
        if md==0:
            re.append(m)
            re.append(a)
        m+=1
    if m**2==n:
        re.append(m)
    return re

n, k = map(int, input().split())
aa = list(map(int, input().split()))
yk = yakusu(sum(aa))
yk.sort(reverse=True)
for y in yk:
    rs=[a%y for a in aa if a%y]
    rs.sort()
    l=0
    sl=rs[0]
    r=len(rs)-1
    sr=y-rs[r]
    while 1:
        if l+1==r:
            if sl==sr and sl<=k:
                print(y)
                exit()
            else:
                break
        if sl>sr:
            r-=1
            sr+=y-rs[r]
            if sr>k:break
        else:
            l+=1
            sl+=rs[l]
            if sl>k:break
print(1)