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: Wrong Answer)

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

public class S
{



    public static void Main()
	{
		/*
        var input = Array.ConvertAll(Console.ReadLine().Split(' '), long.Parse);
        var N = input[0];
        var M = input[1];
        var bridge = new long[M,2];
        for(long i = 0; i < M; i++){
            input = Array.ConvertAll(Console.ReadLine().Split(' '), long.Parse);
            bridge[i,0] = input[0];
            bridge[i,1] = input[1];
        }
        solver(N,M,bridge);
		*/
        solver(8, 7, new long[]{1,7,5,6,8,2,6,5});
    }

    public static void solver(int N, long K, long[] input){

        var sum = input.Sum();

        var divisors = getDivisorList(sum).ToArray();
        Array.Sort(divisors);
        var amari = new long[input.Length];

        // amari list

        var amariPlusPoint = amari.Length-1;
        var moveCount = 0;

        for(int i = divisors.Length-1; i >=0 ; i--){
            for(int k = 0; k < input.Length; k++){
				//Console.WriteLine(k);
                amari[k] = input[k] % divisors[i];
            }
            Array.Sort(amari);
            amariPlusPoint = amari.Length-1;
            moveCount = 0;
            for(int x = 0; x < amari.Length; x++){
                while(amari[x] > 0){
                    amari[x]--;
                    amari[amariPlusPoint]++;
                    moveCount++;
                    if(amari[amariPlusPoint] == divisors[i]){
                        amari[amariPlusPoint] = 0;
                        amariPlusPoint--;
                    }
                }
            }
			
			/*
			Console.Write(divisors[i] + " " + moveCount + " ");
			foreach(var x in amari){
				Console.Write(x + " ");	
			}
			*/
            if(amari.Sum() == 0 && moveCount <= K){
                Console.WriteLine(divisors[i]);
                return;
            }
			
			

        }

        // amari <= K ?


    }


    public static long[] getDivisorList(long num){
        long x = 2;

        var result = new List<long>();
        result.Add(num);
        result.Add(1);
        while(x < num/2){
            if(num % x == 0) result.Add(x);
            x++;
        }
		return result.ToArray();
    }

    class UfTree{
        public long[] parents{get;set;}
        public long[] size{get;set;} //sizeはrootのみが最新のものを保持する

        public UfTree(long n){
            parents = new long[n];
			size = new long[n];
            for(long i = 0; i < n; i++){
                parents[i] = i;
                size[i] = 1;
            } 
        }

        public long getRoot(long x){
            if(parents[x] == x){
                return x;
            }else{
                parents[x] = getRoot(parents[x]);
                return parents[x];
            }
        }

        public long unite(long x, long y){
            var rx = getRoot(x);
            var ry = getRoot(y);
            if(rx == ry){
                return rx;
            } else{
                var sx = getSize(rx);
                var sy = getSize(ry);
                if(sx > sy){
                    size[rx] = sx+sy;
                    parents[ry] = rx;
                    return rx;
                }else{
                    size[ry] = sx+sy;
                    parents[rx] = ry;
                    return ry;
                }
            }
            
        }

        public long getSize(long x){
            return size[getRoot(x)];
        }

        

        public bool isSameTree(long x, long y){
            return getRoot(x) == getRoot(y);
        }

    }

}