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


public class E
{
	public static void Main()
	{
		int N = GetInt();
		int K = GetInt();
		int[] A = Enumerable.Range(0, N).Select(_ => GetInt()).ToArray();

		int sum = A.Sum();
		int sqrt = (int)Math.Sqrt(sum);
		var divisors = new List<int>();
		for(int i = 1; i <= sqrt; i++)
		{
			if(sum % i == 0)
			{
				if(i * i == sum)
					divisors.Add(i);
				else
				{
					divisors.Add(i);
					divisors.Add(sum / i);
				}
			}
		}

		int ans = 1;
		foreach(int divisor in divisors)
		{
			if(sum % divisor != 0)
				continue;

			int[] r = A.Select(x => x % divisor).Where(x => x > 0).ToArray();
			Array.Sort(r);

			int len = r.Length;
			int[] minus = new int[len + 1];
			int[] plus = new int[len + 1];
			for(int i = 0; i < len; i++)
			{
				minus[i + 1] = minus[i] + r[i];
				plus[i + 1] = plus[i] + (divisor - r[i]);
			}

			int numOp = int.MaxValue;
			for(int i = 0; i <= len; i++)
			{
				int tmpMinus = minus[i];
				int tmpPlus = plus[len] - plus[i];
				if(((int)Math.Abs(tmpMinus - tmpPlus)) % divisor != 0)
					continue;

				numOp = Math.Min(numOp, Math.Max(tmpMinus, tmpPlus));				
			}

			if(numOp <= K)
				ans = Math.Max(ans, divisor);
		}

		Console.WriteLine(ans);
	}
}


public static class MyIO
{
	private static string[] args = null;
	private static int num = -1;
	private static int used = -1;

	private static string getArg()
	{
		if(used == num)
		{
			args = Console.ReadLine().Split(' ');
			num = args.Length;
			used = 0;
		}
		return args[used++];
	}

	public static int GetInt(){ return int.Parse(getArg()); }
	public static long GetLong(){ return long.Parse(getArg()); }
	public static double GetDouble(){ return double.Parse(getArg()); }
	public static string GetString(){ return getArg(); }
}