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 Java (OpenJDK 11.0.6) (judge verdict: Time Limit Exceeded)


import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStream;
import java.io.InputStreamReader;
import java.util.List;
import java.util.StringTokenizer;
import java.util.stream.Collectors;
import java.util.stream.IntStream;

public class Main {
    public static void main(final String[] args) {
        final FastScanner scanner = new FastScanner(System.in);
        final int n = scanner.nextInt();
        final int k = scanner.nextInt();

        final List<Integer> list = IntStream.range(0, n)
            .mapToObj(i -> scanner.nextInt())
            .collect(Collectors.toList());

        final int sum = list.stream()
            .mapToInt(Integer::intValue)
            .sum();

        IntStream.range(0, sum)
            .map(i -> sum - i)
            .filter(i -> sum % i == 0)
            .filter(dividable -> {
                final List<Integer> mods = list.stream()
                    .filter(i -> i % dividable != 0)
                    .map(i -> i % dividable)
                    .sorted()
                    .collect(Collectors.toList());
                if (mods.size() == 0) {
                    return true;
                }

                final int[] fromLeft = new int[mods.size()];
                fromLeft[0] = mods.get(0);
                for (int i = 1; i < mods.size(); i++) {
                    fromLeft[i] = fromLeft[i - 1] + mods.get(i);
                }

                final int[] fromRight = new int[mods.size()];
                fromRight[mods.size() - 1] = dividable - mods.get(mods.size() - 1);
                for (int i = mods.size() - 1; i > 0; i--) {
                    fromRight[i - 1] = fromRight[i] + dividable - mods.get(i - 1);
                }

                for (int i = 0; i < mods.size() - 1; i++) {
                    if (fromLeft[i] == fromRight[i + 1]) {
                        return fromLeft[i] <= k;
                    }
                }
                return false;
            })
            .findAny()
            .ifPresent(System.out::println);
    }

    private static class FastScanner {
        private final BufferedReader reader;
        private StringTokenizer tokenizer;

        FastScanner(final InputStream in) {
            reader = new BufferedReader(new InputStreamReader(in));
        }

        String next() {
            if (tokenizer == null || !tokenizer.hasMoreTokens()) {
                try {
                    tokenizer = new StringTokenizer(reader.readLine());
                } catch (final IOException e) {
                    throw new RuntimeException(e);
                }
            }
            return tokenizer.nextToken();
        }

        int nextInt() {
            return Integer.parseInt(next());
        }
    }
}