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: Accepted)

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

    divisors.sort()
    return divisors


def SumArea(r, l, a2):
    return a2[l]-a2[r]

def judge(cand, a, k):
    m_list = [i%cand for i in a]
    m_list.sort()
    p_list = [cand-r for r in m_list]
    
    # 累積和
    b=list(range(1,30))
    m2=[0]
    p2=[0]
    for i in m_list:m2.append(m2[-1]+i)
    for i in p_list:p2.append(p2[-1]+i)
        
    length = len(a)
    for i in range(length):
        m_total  = SumArea(0, i, m2)
        p_total = SumArea(i, length, p2)
        if m_total == p_total and p_total <= k:
            return True
    return False
    
        
n,k = map(int, input().split())
a = list(map(int, input().split()))
pflag = True
# 候補はsum(a)の正約数。
total = sum(a)
length = len(a)
candidate = make_divisors(total)
candidate.sort(reverse = True)
for cand in candidate:
    if judge(cand, a, k):
        print(cand)
        pflag = False
        break
if pflag:
    print(1)