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: Accepted)

using System;
using System.Linq;
using System.Collections.Generic;
using static System.Console;
using System.Text;
using System.IO;
namespace AtCoder_Beginner_Contest_13
{
    class Program
    {
        //配列にする
        static public long[] Sarray() { return ReadLine().Split().Select(long.Parse).ToArray(); }
        //Listにする
        static public List<long> Slist() { return ReadLine().Split().Select(long.Parse).ToList(); }
        static void Main(string[] args)
        {
            SetOut(new StreamWriter(OpenStandardOutput()) { AutoFlush = false });
            Solve();
            Out.Flush();
        }
        static void Solve()
        {
            var NK = Sarray();
            var N = NK[0];
            var K = NK[1];
            var Ai = Slist();

            var sum = Ai.Sum();
            var max = Ai.Max();
            if (0 == max)
            {
                WriteLine(0);
                return;
            }
            var dc = yaku(sum).OrderByDescending(e => e.Key);
            foreach (var pk in dc)
            {
                Ai.Sort((l, r) => (int)(r % pk.Key - l % pk.Key));
                var b = 0;
                var e = (int)N - 1;
                var ln = Ai[b] % pk.Key;
                var rn = Ai[e] % pk.Key;
                var ope = 0L;
                while (b < e)
                {
                    if (K < ope)
                        break;
                    if (pk.Key == ln + rn)
                    {
                        ope += rn;
                        ln = Ai[++b] % pk.Key;
                        rn = Ai[--e] % pk.Key; ;
                        if (e <= b)
                        {
                            if (ope <= K)
                            {
                                WriteLine(pk.Key);
                                return;
                            }
                        }
                    }
                    else if (pk.Key < ln + rn)
                    {
                        ope += pk.Key - ln;
                        rn = ln + rn - pk.Key;
                        ln = Ai[++b] % pk.Key;
                    }
                    else
                    {
                        ope += rn;
                        ln += rn;
                        rn = Ai[--e] % pk.Key;
                    }
                }
            }
            WriteLine(1);
        }
        static Dictionary<long, long> yaku(long n)
        {
            var ret = new Dictionary<long, long>();
            for (long i = 1; i * i <= n; i++)
            {
                if (n % i == 0)
                {
                    ret[i] = 1;
                    ret[n / i] = 1;
                }
            }
            return ret;
        }
    }
}