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)

//★要ＴＬＥ回避
using System;
using System.IO;
using System.Collections.Generic;
using System.Text.RegularExpressions;
class Program
{
    static string result;
static long goukei=0;
static long goukei2=0;
static int kosuu;
static long kaisuu;
static int[] B;
static int[] C;
static long y=0;
    static void Main()
    {
        string[] tmp2 = Console.ReadLine().Split(' ');
kosuu=int.Parse(tmp2[0]);
kaisuu=long.Parse(tmp2[1]);   
string[] tmp = Console.ReadLine().Split(' ');
int[] A=new int[kosuu];
B=new int[kosuu];
C=new int[kosuu];
for(int i=0;i<kosuu;i++){A[i]=int.Parse(tmp[i]);}
long x=0;
for(int i=0;i<kosuu;i++){x+=A[i];}
//xは総合計。
//yは求める数

long s=x;
for(int j=0;j<kosuu;j++){B[j]=A[j]%(int)s;}
for(int j=0;j<kosuu;j++){C[j]=(int)s-A[j]%(int)s;}
//チョイスしたBの合計と残りのCの合計の絶対値がkaisuu以下になる組み合わせがあるか探査。あればiをyに保存
Combi(0,s);
for(long i=x/2;i>0;i--)
{if(x%i>0){continue;}//割り切れなければ×
for(int j=0;j<kosuu;j++){B[j]=A[j]%(int)i;}
for(int j=0;j<kosuu;j++){C[j]=(int)i-A[j]%(int)i;}
//チョイスしたBの合計と残りのCの合計の絶対値がkaisuu以下になる組み合わせがあるか探査。あればiをyに保存
Combi(0,i);
}

result=y.ToString();
        Console.WriteLine(result);
        
    }

/*
//再帰関数と同機能のループ
for(int i=0;i<kosuu;i++)
{
//現状の変数状態を保存


//


}
*/




//いかに再帰回数を削減するか（ループ化の方がいいかも？）
static void Combi(int junban,long i)//(チェックする配列番号,i(仮定している公約数))
{
if(junban==kosuu){if(goukei<=kaisuu && goukei2<=kaisuu && goukei==goukei2){y=i;}return;}
goukei2+=C[junban];
Combi(junban+1,i);//引く場合
goukei2-=C[junban];
goukei+=B[junban];//合計値に追加
Combi(junban+1,i);//足す場合
goukei-=B[junban];//終わったら足した分を戻す
}
}