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: Accepted)

import java.util.*;
import java.io.*;
import java.nio.charset.StandardCharsets;

public class Main {

	public static void main(String[] args) throws IOException {
		InputStreamReader reader = new InputStreamReader(System.in, StandardCharsets.UTF_8);
		BufferedReader in = new BufferedReader(reader);

		String[] tokens = in.readLine().split(" ");
		int N = Integer.parseInt(tokens[0]);
		long K = Long.parseLong(tokens[1]);
		tokens = in.readLine().split(" ");
		long[] A = new long[N];
		for (int i = 0; i < N; ++i) {
			A[i] = Long.parseLong(tokens[i]);
		}
		long sum = 0;
		for (int i = 0; i < N; ++i) {
			sum += A[i];
		}
		List<Long> yakusuu = getYakusuu(sum);
		long result = 1;
		for (int i = 0; i < yakusuu.size(); ++i) {
			if (isEnable(A, K, yakusuu.get(i))) {
				result = yakusuu.get(i);
				break;
			}
		}
		System.out.println(result);
	}

	static boolean isEnable(long[] A, long K, long target) {
		List<Long> a = new ArrayList<>();
		for (int i = 0; i < A.length; ++i) {
			a.add(A[i] % target);
		}
		Collections.sort(a);
		
		Deque<Long> deque= new ArrayDeque<>(a.size());
		deque.addAll(a);

		long count = 0;
		long diff = 0;
		while(!deque.isEmpty()) {
			if(deque.getLast()+diff >= target) {
				diff -= target - deque.pollLast();
			}else {
				long val = deque.pollFirst();
				count += val;
				diff += val;
			}
			if(count>K) {
				return false;
			}
		}
		return true;
	}

	static List<Long> getYakusuu(long value) {
		Deque<Long> small = new ArrayDeque<>();
		Deque<Long> large = new ArrayDeque<>();
		for (long v = 1; v <= 1 + (long) Math.floor(Math.sqrt((double) value)); ++v) {
			if (value % v == 0) {
				small.addFirst(v);
				large.addLast(value / v);
			}
		}
		List<Long> result = new ArrayList<>(small.size() + large.size());
		result.addAll(large);
		result.addAll(small);
		return result;
	}
}