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;
using System.Collections.Generic;
using static Assistant.Input;
using static Assistant.Debug;
using System.Linq;
using Assistant;

namespace ABC136E
{
    class Program
    {
        static void Main(string[] args)
        {
            var N = RInt;
            var K = RInt;
            var A = RLongs;
            long S = A.Sum();

            var fact = GetDivisors(S).ToArray();
            fact = fact.OrderByDescending(x => x).ToArray();

            for (int i = 0; i < fact.Length; i++)
            {
                long cand = fact[i];
                long rem = S / cand;

                var Amod = A.Select(y => y % cand).OrderBy(y => y).ToArray();
                long[] combF = new long[N]; combF[0] = Amod[0];
                long[] combL = new long[N]; combL[N - 1] = cand - Amod[N - 1];
                for (int j = 0; j < N - 1; j++)
                {
                    combF[j + 1] = combF[j] + Amod[j + 1];
                    combL[N - j - 2] = combL[N - j - 1] + cand - Amod[N - j - 2];
                }

                long count = int.MaxValue;
                for (int j = 0; j < N - 1; j++)
                {
                    long c = Math.Abs(combL[j + 1] - combF[j]);
                    if (c % cand != 0) continue;

                    c = combF[j] + combL[j + 1];
                    count = Math.Min(count, c);
                }
                if (count < 2 * K)
                {
                    Console.WriteLine(cand);
                    break;
                }
            }

        }

        /// <summary>
        /// 約数の列挙
        /// </summary>
        /// <param name="num"></param>
        /// <returns></returns>
        public static IEnumerable<long> GetDivisors(long num)
        {
            if (num < 1) yield break;

            for (long i = 1; i * i <= num; i++)
            {
                if (num % i == 0)
                {
                    yield return i;
                    if (i * i != num) yield return (num / i);
                }
            }
        }
    }


}

namespace Assistant
{
    static class Input
    {
        static List<string> line = new List<string>();
        static int index = 0;
        static String RNext()
        {
            if (line.Count <= index) line.AddRange(Console.ReadLine().Split());
            return line[index++];
        }
        public static int RInt => int.Parse(RNext());
        public static long RLong => long.Parse(RNext());
        public static int[] RInts => Console.ReadLine().Split().Select(int.Parse).ToArray();
        public static long[] RLongs => Console.ReadLine().Split().Select(long.Parse).ToArray();
        public static string RString => RNext();
        //以下未テスト
        public static int[] RIntsC(int c) => Enumerable.Repeat(0, c).Select(x => int.Parse(RNext())).ToArray();
        public static long[] RLongsC(int c) => Enumerable.Repeat(0, c).Select(x => long.Parse(RNext())).ToArray();
        public static char[][] RMap(int h) => Enumerable.Repeat(0, h).Select(x => Console.ReadLine().ToCharArray()).ToArray();
    }

    static class Debug
    {
        static public void Draw2D<T>(T[,] map, int mode = 1)
        {
#if DEBUG
            int W = map.GetLength(0);
            int H = map.GetLength(1);

            string[,] map2 = new string[W + 1, H + 1];
            for (int i = 0; i < W + 1; i++)
            {
                for (int j = 0; j < H + 1; j++)
                {
                    if (i == 0 && j == 0) map2[i, j] = 0.ToString();
                    else if (i == 0) map2[i, j] = (j - 1).ToString();
                    else if (j == 0) map2[i, j] = (i - 1).ToString();
                    else map2[i, j] = map[i - 1, j - 1].ToString();
                }
            }

            for (int i = 0; i < W + 1; i++)
            {
                for (int j = 0; j < H + 1; j++)
                {
                    if (mode == 0) Console.Write(map2[i, j].Last());
                    if (mode == 1) Console.Write(map2[i, j] + " ");
                }
                Console.WriteLine();
            }
            Console.WriteLine();
#endif
        }

        public static void Draw1D<T>(T[] array, int mode = 0)
        {
#if DEBUG
            Console.WriteLine(string.Join(" ", array));
#endif
        }
    }
}