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.Collections.Generic;
using System.Linq;

namespace ABC136E
{
    class Program
    {
        static void Main(string[] args)
        {
            long[] NK = CIN.LongArray();
            long N = NK[0];
            long K = NK[1];
            long[] A = CIN.LongArray();

            long Sum = A.Sum();
            List<long> Factors = new List<long>();
            var root = Math.Sqrt(Sum);
            for (int i = 1; i <= root; i++)
            {
                if (Sum % i == 0)
                {
                    Factors.Add(i);
                }
            }

            int fs = Factors.Count;
            for (int i = 0; i < fs; i++)
            {
                Factors.Add(Sum / Factors[i]);
            }

            long ans = 1;
            foreach (long factor in Factors)
            {
                if (factor == 1)
                {
                    continue;
                }

                long[] AA = new long[N];
                for (int i = 0; i < N; i++)
                {
                    AA[i] = A[i] % factor;
                }
                Array.Sort(AA);

                long[] Upp = new long[N];
                long[] Low = new long[N];
                Upp[0] = AA[0];
                Low[N - 1] = factor - AA[N - 1];
                for (int i = 1; i < N; i++)
                {
                    Upp[i] = Upp[i - 1] + AA[i];
                }

                for (int i = (int)N - 2; i >= 0; i--)
                {
                    Low[i] = Low[i + 1] + (factor - AA[i]);
                }

                for (int i = 0; i < N - 1; i++)
                {
                    if (Upp[i] == Low[i + 1] && Upp[i] <= K)
                    {
                        ans = Math.Max(factor, ans);
                        break;
                    }

                    if (Low[i] == Upp[i + 1] && Upp[i] <= K)
                    {
                        ans = Math.Max(factor, ans);
                        break;
                    }
                }
            }
            
            Console.WriteLine(ans);
        }
    }

    internal static class CIN
    {
        public static string String()
        {
            return Console.ReadLine();
        }

        public static int Int()
        {
            return int.Parse(Console.ReadLine());
        }

        public static int[] IntArray()
        {
            return Console.ReadLine().Split(' ').Select(int.Parse).ToArray();
        }

        public static long Long()
        {
            return long.Parse(Console.ReadLine());
        }

        public static long[] LongArray()
        {
            return Console.ReadLine().Split(' ').Select(long.Parse).ToArray();
        }

        public static double[] DoubleArray()
        {
            return Console.ReadLine().Split(' ').Select(double.Parse).ToArray();
        }
    }
}