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 C# (Mono 4.6.2.0) (judge verdict: Time Limit Exceeded)

using System;
using System.Linq;
using System.Text;
using System.Collections.Generic;

public class Hello{
    
    private static int N;
    private static int K;
    
    public static void Main(){
        var line = Console.ReadLine().Split(' ');
        N = int.Parse(line[0]);
        K = int.Parse(line[1]);
        var sequence = Console.ReadLine().Split(' ').Select(int.Parse).ToArray();
        int sum = sequence.Sum();
        var candidate = new List<int>();
        for(int i=1;i<=sum;i++){
            if(sum%i==0){
                candidate.Add(i);
            }
        }
        int ans = 0;
        foreach(int cand in candidate.OrderByDescending(x => x)){
            if(PlusOrMinus(0,sequence.Select(x => x%cand).Where(x => x!=0).ToArray(),0,0,cand)){
                ans = cand;
                break;
            }
        }
        Console.WriteLine(ans);
    }
    private static bool PlusOrMinus(int index, int[] arr, int plus, int minus, int cand){
        if(arr.Length<=0){
            return true;
        }
        if(index==arr.Length){
            if(plus==minus && plus<=K){
                return true;
            }else{
                return false;
            }
        }
        return PlusOrMinus(index+1,arr,plus+arr[index],minus,cand) || PlusOrMinus(index+1,arr,plus,minus+cand-arr[index],cand);
    }
}