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'
class Integer
  def divisor_list
    return [1] if self == 1
    Prime.prime_division(self).map do |e|
      Array.new(e[1]+1).map.with_index do |element, i|
        e[0]**i
      end
    end.inject{|p,q| p.product(q)}.map do |a|
      [a].flatten.inject(&:*)
    end.sort.reverse
  end
end

n, k = gets.split.map(&:to_i)
as = gets.split.map(&:to_i)
sum = as.inject(:+)

sum.divisor_list.each do |divisor|
  plus_neccesities = []
  minus_neccesities = []
  as.each do |a|
    minus_neccesity = divisor % a
    plus_neccesity = divisor - minus_neccesity
    if minus_neccesity < plus_neccesity
      minus_neccesities << minus_neccesity
    else
      plus_neccesities << plus_neccesity
    end
  end
    minus_neccesities << 0
    plus_neccesities << 0
    sum_of_minus_neccesity = minus_neccesities.inject(:+)
    sum_of_plus_neccesity = plus_neccesities.inject(:+)
    able_times = divisor / 2 - 1

    if sum_of_minus_neccesity > sum_of_plus_neccesity
      sum_of_minus_neccesity -= 1
      sum_of_plus_neccesity += divisor - 1
    elsif sum_of_plus_neccesity > sum_of_minus_neccesity
      sum_of_plus_neccesity -= 1
      sum_of_minus_neccesity += divisor - 1
    end

    if [sum_of_minus_neccesity, sum_of_plus_neccesity].max - able_times < k
      puts divisor
      exit
    end
  
end