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;


public class Contest
{
	public static void Main()
	{

/*
		int n = a[0];
		int[][] e = new int[n][];
		for(int i=0; i<n; i++) MyIO.ReadArray(out e[i]);

		int[] a;
		MyIO.ReadArray(out a);

        long a,b,c,d,e,f;
        MyIO.ReadMulti(out a, out b, out c, out d, out e, out f);

		MyIO.WriteArray(ans);
*/

        int n,k;
		MyIO.ReadMulti(out n, out k);
		int[] a;
		MyIO.ReadArray(out a);

		int sum = 0;
		List<int> dn = new List<int>();

		for(int i=0;i<n; i++) sum += a[i];
		for(int i=1;i<(int)Math.Sqrt(sum)+1; i++) {
			if(sum%i==0)dn.Add(i);
		}
		int dc = dn.Count;
		for(int i=0; i<dc; i++){
			if(dn[dc-i-1]!=sum/dn[dc-i-1])dn.Add(sum/dn[dc-i-1]);
		}
		int[] da = dn.ToArray();

		int ans = 0;
		for(int i=da.Length-1; 0<=i; i--){
			int sm = 0;
			int[] am = new int[n];
			for(int j=0; j<n; j++) {
				am[j] = a[j]%da[i];
				sm += am[j];
			}
			Array.Sort(am);
			int fi = sm / da[i];
			int ks = 0;
			for(int j=0; j<n-fi; j++) ks += am[j];
			if(ks<=k){
				ans = da[i];
				break;
			}
		}

		Console.WriteLine(ans);
	}


}

public static class MyIO
{
	private static bool eq<T, U>() => typeof(T).Equals(typeof(U));
	private static T ct<T, U>(U a) => (T)Convert.ChangeType(a, typeof(T));
	private static T cv<T>(string s) => eq<T, int>()    ? ct<T, int>(int.Parse(s))
	                                  : eq<T, long>()   ? ct<T, long>(long.Parse(s))
	                                  : eq<T, double>() ? ct<T, double>(double.Parse(s))
	                                  : eq<T, char>()   ? ct<T, char>(s[0])
	                                                    : ct<T, string>(s);
	public static string[] ReadArray<T>(out T[] a) {		
		var ar = Console.ReadLine().Split(new char[] {' '}, StringSplitOptions.RemoveEmptyEntries); 												
		a = System.Array.ConvertAll(ar, new Converter<string,T>(cv<T>));
		return ar;
	}										
	public static string[] ReadMulti<T>(out T a) {
		var ar = Console.ReadLine().Split(new char[] {' '}, StringSplitOptions.RemoveEmptyEntries); 
		a = cv<T>(ar[0]);
		return ar;
	}
	public static string[] ReadMulti<T, U>(out T a, out U b) {
		var ar = ReadMulti<T>(out a); 
        b = cv<U>(ar[1]);
		return ar;
	}
	public static string[] ReadMulti<T, U, V>(out T a, out U b, out V c) {
		var ar = ReadMulti<T, U>(out a, out b); 
        c = cv<V>(ar[2]);
		return ar;
	}
	public static string[] ReadMulti<T, U, V, W>(out T a, out U b, out V c, out W d) {
		var ar = ReadMulti<T, U, V>(out a, out b, out c); 
        d = cv<W>(ar[3]);
		return ar;
	}
	public static string[] ReadMulti<T, U, V, W, X>(out T a, out U b, out V c, out W d, out X e) {
		var ar = ReadMulti<T, U, V, W>(out a, out b, out c, out d); 
        e = cv<X>(ar[4]);
		return ar;
	}
	public static string[] ReadMulti<T, U, V, W, X, Y>(out T a, out U b, out V c, out W d, out X e, out Y f) {
		var ar = ReadMulti<T, U, V, W, X>(out a, out b, out c, out d, out e); 
        f = cv<Y>(ar[5]);
		return ar;
	}

	public static void WriteArray<T>(T[] a) {
		Console.WriteLine(string.Join(" ", Array.ConvertAll(a, x => x.ToString())));
	}


}