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: Accepted)

N,K=gets.split.map &:to_i
as=gets.split.map &:to_i

class Integer
    def divisor_list
        return [1] if self == 1
        (1..Math.sqrt(self).to_i).select do |i|
            self%i == 0
        end.map {|e| [e, self/e]}.flatten.uniq.sort
    end
end

cand=as.inject(:+).divisor_list
cand.reverse.each do |m|
    as_mod=as.map{|i|m-i%m}.sort.reverse
    as_mod.delete(0)
    if as_mod.nil?
        puts m
        exit 0
    end
    rcnt=as_mod.inject(:+)
    lcnt=0
    for i in 0..as_mod.size-2
        rcnt+=-as_mod[i]
        lcnt+=-(m-as_mod[i])
        if rcnt % m == -lcnt % m && [rcnt,-lcnt].max <= K
            puts m
            exit 0
        end
    end
end