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)

de = false
require 'prime'
n,k = gets.chomp.split().map(&:to_i)
a = gets.chomp.split().map(&:to_i)
a.sort!
a.reverse!

def divisors(n)
  n.prime_division.inject([1]) do |ary, (p, e)|
    (0..e).map{ |e1| p ** e1 }.product(ary).map{ |a, b| a * b }
  end.sort
end

divs = divisors(a.inject(:+)).reverse
p divs if de
divs.each do |divisor|
  divided = a.map{|e| e%divisor}.sort.select{|e| e>0}
  p [divisor, 'divided', divided] if de
  rui = []
  divided.each do |e|
    rui.push (rui.last || 0) + e
  end
  rui2 = []
  divided.reverse_each do |e|
    rui2.push (rui2.last || 0) + divisor - e
  end
  rui2.reverse!

  minimum = Float::INFINITY
  size = divided.size
  p ['rui', rui] if de
  p ['rui2', rui2] if de
  (size-1).times do |i|
    diff = [rui[i], rui2[i+1]].max
    minimum = diff if minimum > diff
  end

  p ['minim', minimum] if de
  if minimum <= k
    puts divisor
    break
  end
end