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 Java8 (OpenJDK 1.8.0) (judge verdict: Time Limit Exceeded)

//package abc136;
import java.io.*;
import java.util.*;

public class Main{
	// ------------------------
	
	public static void main(String[] args) {
		MyScanner sc = new MyScanner();
		out = new PrintWriter(new BufferedOutputStream(System.out));
		// ------------------------
		int n = sc.nextInt(), k = sc.nextInt();
		long sum = 0;
		long[] a = new long[n];
		for(int i = 0; i < n; i++)
			sum += a[i] = sc.nextInt();
		long ans = 1;
		for(int i = 1; i * i <= sum; i++)
			if(sum % i == 0) {
				if(i > ans && works(a, k, i))
					ans = i;
				if(sum / i > ans && works(a, k, sum / i))
					ans = sum / i;	
			}
		out.println(ans);
		// ------------------------
		out.close();
	}
	//------------------------
	public static boolean works(long[] a, int k, long x) {
		long[] b = new long[a.length];
		for(int i = 0; i < b.length; i++)
			b[i] = a[i] % x;
		Arrays.sort(b);
		int add = b.length - 1, sub = 0;
		for(int i = 0; i < k; i++) {
			while(add >= 0 && b[add] == x)
				add--;
			while(sub < b.length && b[sub] == 0)
				sub++;
			if(add <= sub)
				break;
			b[add]++;
			b[sub]--;
		}
		for(long i : b)
			if(i % x != 0)
				return false;
		return true;
	}
	public static PrintWriter out;
	public static class MyScanner {
		BufferedReader br;
		StringTokenizer st;
		public MyScanner() {
			br = new BufferedReader(new InputStreamReader(System.in));
		}
		String next() {
			while (st == null || !st.hasMoreElements()) {
				try {
					st = new StringTokenizer(br.readLine());
				} catch (IOException e) {
					e.printStackTrace();
				}
			}
			return st.nextToken();
		}
		int nextInt() {
			return Integer.parseInt(next());
		}
		long nextLong() {
			return Long.parseLong(next());
		}
		double nextDouble() {
			return Double.parseDouble(next());
		}
		String nextLine() {
			String str = "";
			try {
				str = br.readLine();
			} catch (IOException e) {
				e.printStackTrace();
			}
			return str;
		}
	}
}