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)

class Integer
  def divisor_list
    require 'prime'

    return [] if self <= 0
    return [1] if self == 1

    prime_division.map.with_index { |(base, k), i|
      s = i.zero? ? 0 : 1
      (s..k).map { |n| base ** n }
    }.inject { |res, e| res + res.flat_map { |t| e.map { |v| t * v } } }.sort
  end
end

N, K = gets.split.map(&:to_i)
A = gets.split.map(&:to_i)

def f(x)
  upDP = Array.new(N) { Hash.new(false) }
  downDP = Array.new(N) { Hash.new(false) }

  A.each_with_index do |a, i|
    down = a % x
    up = x - down

    limit = 2 * K

    if i.zero?
      upDP[0][up] = up
      downDP[0][-down] = down
    else
      up_keys = upDP[i - 1].keys

      up_keys.each do |k|
        cnt = upDP[i - 1][k]

        upDP[i][k + up] = cnt + up if cnt + up <= limit
        downDP[i][k - down] = cnt + down if cnt + down <= limit
      end

      down_keys = downDP[i - 1].keys

      down_keys.each do |k|
        cnt = downDP[i - 1][k]

        upDP[i][k + up] = cnt + up if cnt + up <= limit
        downDP[i][k - down] = cnt + down if cnt + down <= limit
      end
    end

    # p [x, downDP[i], upDP[i]]
  end

  upDP[N - 1][0] || downDP[N - 1][0]
end

sum = A.inject(:+)
ans = 1

sum.divisor_list.each do |x|
  ans = x if f(x) && ans < x
end

puts ans