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: Accepted)

N,K=list(map(int,input().split()))
A=list(map(int,input().split()))
S=sum(A)
facts=set()
for i in range(1,int(S**0.5)+1):
    if S%i==0:
        facts.add(i)
        facts.add(S//i)
for f in sorted(facts,key=lambda x:-x):
    rs = [i%f for i in A]
    rs2=sorted(rs)
    rsr = [f-r for r in rs2]
    if f-rs2[-1]>K and rs2[-1]>K:
        continue
    cum=0
    ruiseki=[0]*(N)
    tmp=0
    for i,a in enumerate(rs2):
        cum+=a
        ruiseki[i]=cum
    for i in range(N-1):
        s_l=ruiseki[i]
        s_h = f*(N-i-1)-(ruiseki[-1]-s_l)
        if max(s_l,s_h)<=K:
            print(f)
            break
    else:
        continue
    break