Problem:
Score : 600 points

Problem Statement

Snuke has a blackboard and a set S consisting of N integers.
The i-th element in S is S_i.

He wrote an integer X on the blackboard, then performed the following operation N times:

Choose one element from S and remove it.

Let x be the number written on the blackboard now, and y be the integer removed from S. Replace the number on the blackboard with x \bmod {y}.

There are N! possible orders in which the elements are removed from S.
For each of them, find the number that would be written on the blackboard after the N operations, and compute the sum of all those N! numbers modulo 10^{9}+7.

Constraints

All values in input are integers.

1 \leq N \leq 200

1 \leq S_i, X \leq 10^{5}

S_i are pairwise distinct.

Input

Input is given from Standard Input in the following format:

N X
S_1 S_2 \ldots S_{N}

Output

Print the answer.

Sample Input 1

2 19
3 7

Sample Output 1

3

There are two possible orders in which we remove the numbers from S.

If we remove 3 and 7 in this order, the number on the blackboard changes as follows: 19 \rightarrow 1 \rightarrow 1.

If we remove 7 and 3 in this order, the number on the blackboard changes as follows: 19 \rightarrow 5 \rightarrow 2.

The output should be the sum of these: 3.

Sample Input 2

5 82
22 11 6 5 13

Sample Output 2

288

Sample Input 3

10 100000
50000 50001 50002 50003 50004 50005 50006 50007 50008 50009

Sample Output 3

279669259

Be sure to compute the sum modulo 10^{9}+7.

Solution in C# (Mono 4.6.2.0) (judge verdict: Accepted)

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

namespace エクサウィザーズ2019D
{
    class Program
    {
        static void Main(string[] args)
        {
            int N = ReadInt;
            int X = ReadInt;
            int mod = 1000000007;

            List<int> S = ReadInts.OrderBy(x => -x).ToList();

            int xMax = Math.Max(S.Max(), X) + 1;
            long[,] dp = new long[N + 1, xMax + 1];
            dp[0, X] = 1;

            for (int i = 0; i < N; i++)
            {
                for (int j = 1; j < xMax; j++)
                {
                    dp[i + 1, j % S[i]] += dp[i, j];
                    dp[i + 1, j % S[i]] %= mod;

                    dp[i + 1, j] += dp[i, j] * (N - i - 1) % mod;
                    dp[i + 1, j] %= mod;

                }
                //DrawMapforDebug(dp);
            }

            long ans = 0;
            for (int x = 0; x < xMax; x++)
            {
                ans += dp[N, x] * x % mod;
                ans %= mod;
            }

            Console.WriteLine(ans);
        }


        /// <summary>
        /// 二次元配列の可視化メソッド
        /// </summary>
        /// <param name="map"></param>
        static public void DrawMapforDebug(int[,] map, int mode = 1)
        {
            int W = map.GetLength(0);
            int H = map.GetLength(1);

            int[,] map2 = new int[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;
                    else if (i == 0) map2[i, j] = j - 1;
                    else if (j == 0) map2[i, j] = i - 1;
                    else map2[i, j] = map[i - 1, j - 1];
                }
            }

            for (int i = 0; i < W + 1; i++)
            {
                for (int j = 0; j < H + 1; j++)
                {
                    if (mode == 0) Console.Write(map2[i, j] % 10);
                    if (mode == 1) Console.Write(map2[i, j] + " ");
                }
                Console.WriteLine();
            }
            Console.WriteLine();
        }
    }


}

namespace Assistant
{
    static class Input
    {
        static List<string> line = new List<string>();
        static int index = 0;
        static String ReadNext()
        {
            if (line.Count <= index) line.AddRange(Console.ReadLine().Split());
            return line[index++];
        }
        public static int ReadInt => int.Parse(ReadNext());
        public static long ReadLong => long.Parse(ReadNext());
        public static int[] ReadInts => Console.ReadLine().Split().Select(int.Parse).ToArray();
        public static long[] NextLongs => Console.ReadLine().Split().Select(long.Parse).ToArray();
    }

    static class Debug
    {
        //drawmap
        //drawarray
    }
}