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 Ruby (2.3.3) (judge verdict: Wrong Answer)

require 'prime'

# 約数リスト
def divisors(n)
  n.prime_division.inject([1]) do |array, (k, v)|
    (0..v).map{ |i| k ** i }.product(array).map{ |a, b| a * b }
  end.sort
end

n, k = gets.split.map(&:to_i)
nums = gets.split.map(&:to_i)
sum = nums.inject(:+)
divisors = divisors(sum).reverse
divisors.each do |div|
  mods = nums.map { |n| n % div }.sort
  cumulative_mods = mods.inject([]) { |res, mod| res << (res.last || 0) + mod }
  cumulative_revs = mods.reverse.inject([]) { |res, mod| res << (res.last || 0) -(mod - div) }
  (n - 1).times do |index|
    rev_index = n - index - 2
    if cumulative_mods[index] == cumulative_revs[rev_index]
      break if cumulative_mods[index] > k
      puts div
      exit
    end
  end
end