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.*;

public class Main {
    public static void main(String[] args) {
        FastScanner scanner = new FastScanner();
        PrintWriter out = new PrintWriter(System.out);
        int n = scanner.nextInt();
        int k = scanner.nextInt();
        int sum = 0;
        int[] arr = new int[n];
        int ans = 1;
        for(int i = 0; i < n; i++) {
            arr[i] = scanner.nextInt();
            sum += arr[i];
        }
        ArrayList<Integer> factors = new ArrayList<>();
        factors.add(1);factors.add(sum);
        for(int i = 2; i*i<=sum; i++) {
            if (sum % i == 0) {
                if (sum/i != i) factors.add(sum/i);
                factors.add(i);
            }
        }
        for(int i =0; i < factors.size(); i++) {
            int curfactor = factors.get(i);
            TreeMap<Integer, Integer> mapping = new TreeMap<>();
            long cost = 0;
            long used = 0;
            for(int j = 0; j < n; j++) {
                int cur = arr[j] % curfactor;
                if (!mapping.containsKey(cur)) mapping.put(cur, 0);
                mapping.put(cur, mapping.get(cur) + 1);
                cost += curfactor - cur;
            }
            while(!mapping.isEmpty()) {
                int minKey = mapping.firstKey();
                mapping.put(minKey, mapping.get(minKey)-1);
                if (mapping.get(minKey) == 0) mapping.remove(minKey);
                used += minKey;
                cost -= curfactor - minKey;
                //this is guaranteed to trigger since the total mod adds to zero
                if (cost == used && used <= k) ans = Math.max(ans, curfactor);
            }
        }
        out.println(ans);
        out.flush();
    }
    
    public static class FastScanner {
        BufferedReader br;
        StringTokenizer st;
        
        public FastScanner(Reader in) {
            br = new BufferedReader(in);
        }
        
        public FastScanner() {
            this(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 readNextLine() {
            String str = "";
            try {
                str = br.readLine();
            } catch (IOException e) {
                e.printStackTrace();
            }
            return str;
        }
    }
}