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: Wrong Answer)

using System;
class Program
{
    static void Main(string[] args)
    {
        string[] input = Console.ReadLine().Split(' ');
        int N = int.Parse(input[0]);
        int K = int.Parse(input[1]);
        string[] As = Console.ReadLine().Split(' ');
        int[] A = new int[N];
        int sum = 0;
        for (var i = 0; i < N; i++)
        {
            A[i] = int.Parse(As[i]);
            sum += A[i];
        }
        int ans = 1;
        for (var d = sum; d <= sum; d++)
        {
            int[] ct_mod = new int[d];
            for (var i = 0; i < N; i++)
            {
                ct_mod[A[i] % d]++;
            }
            int diff_sum_p = 0;
            int diff_sum_m = 0;
            for (var i = 1; i <= (d - 1) / 2; i++)
            {
                diff_sum_m += ct_mod[i] * i;
                diff_sum_p += ct_mod[d - i] * i;
            }
            if (d % 2 == 0)
            {
                var i = d / 2;
                diff_sum_m += ct_mod[i] * i;
            }
            if (diff_sum_p + diff_sum_p > K * 2) continue;
            int dc = (diff_sum_p - diff_sum_m) / d;
            if (dc < 0)
            {
                for (var i = d / 2; i > 0 && dc != 0; i--)
                {
                    int mov = Math.Min(-dc, ct_mod[i]);
                    diff_sum_m -= i * mov;
                    diff_sum_p += (d - i) * mov;
                    dc += mov;
                }
            }
            if (dc > 0)
            {
                for (var i = (d - 1) / 2; i > 0 && dc != 0; i--)
                {
                    int mov = Math.Min(dc, ct_mod[d - i]);
                    diff_sum_p -= i * mov;
                    diff_sum_m += (d - i) * mov;
                    dc -= mov;
                }
            }
            if (dc == 0 && diff_sum_p <= K)
            {
                ans = d;
                break;
            }
        }
        if (ans == 1)
        {
            int rt = (int)Math.Sqrt(sum) + 1;
            for (var d = rt; d >= 1; d--)
            {
                if (sum % d == 0)
                {
                    int[] ct_mod = new int[d];
                    for (var i = 0; i < N; i++)
                    {
                        ct_mod[A[i] % d]++;
                    }
                    int diff_sum_p = 0;
                    int diff_sum_m = 0;
                    for (var i = 1; i <= (d - 1) / 2; i++)
                    {
                        diff_sum_m += ct_mod[i] * i;
                        diff_sum_p += ct_mod[d - i] * i;
                    }
                    if (d % 2 == 0)
                    {
                        var i = d / 2;
                        diff_sum_m += ct_mod[i] * i;
                    }
                    if (diff_sum_p + diff_sum_p > K * 2) continue;
                    int dc = (diff_sum_p - diff_sum_m) / d;
                    if (dc < 0)
                    {
                        for (var i = d / 2; i > 0 && dc != 0; i--)
                        {
                            int mov = Math.Min(-dc, ct_mod[i]);
                            diff_sum_m -= i * mov;
                            diff_sum_p += (d - i) * mov;
                            dc += mov;
                        }
                    }
                    if (dc > 0)
                    {
                        for (var i = (d - 1) / 2; i > 0 && dc != 0; i--)
                        {
                            int mov = Math.Min(dc, ct_mod[d - i]);
                            diff_sum_p -= i * mov;
                            diff_sum_m += (d - i) * mov;
                            dc -= mov;
                        }
                    }
                    if (dc == 0 && diff_sum_p <= K)
                    {
                        ans = d;
                        break;
                    }
                }
            }
        }
        Console.WriteLine(ans);
    }
}