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"
n,k = gets.chomp.split.collect{ |item| item.to_i}
alist = gets.chomp.split.collect{ |item| item.to_i}
max = alist.inject(:+)
gcd = max.prime_division
alist.sort!
ald = true
if k >= (max - alist.max)
  puts max
  ald = false
end

def check(num,list,maxk)
  cost = 0
  list.each do |li|
    cost += li%num
  end
  if maxk >= cost/2
    return true
  else
    return false
  end
end

clist = []
gcd.each do |set|
  set[1].times { clist << set[0]}
end
result = 1
if ald
  testlist = []
  1.upto(clist.size) do |nn|
    clist.combination(nn).each do |testset|
      test = testset.inject(:*)
      testlist << test if test != max
    end
  end
  testlist.sort.each do |test|
    if check(test,alist,k)
      result = test
    end
  end
  puts result
end