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.IO;
using System.Linq;
using System.Collections;
using System.Collections.Generic;
using System.Text;
using System.Numerics;
using System.Threading.Tasks;
using System.Text.RegularExpressions;
using static System.Math;
using Debug = System.Diagnostics.Debug;
using LayoutKind = System.Runtime.InteropServices.LayoutKind;
using MethodImplOptions = System.Runtime.CompilerServices.MethodImplOptions;
using MethodImplAttribute = System.Runtime.CompilerServices.MethodImplAttribute;


static class P
{
    static void Main()
    {
        var nx = Console.ReadLine().Split().Select(int.Parse).ToArray();
        long X = nx[1];
        //ちゅーごくじょーよてーり
        //オーダーによってSの終端が決定する
        //まず影響するのは降順のみなことは自明に分かって、
        //で、「次」を選ぶことができる順番は
        //「確率」「期待値」
        //aを取り除いた場合、その後に適用できるのはそれ以前全て。

        List<int> s = Console.ReadLine().Split().Select(int.Parse).OrderByDescending(x => x).ToList();
        ModInt[] dp = new ModInt[X + 1];
        dp[X] = 1;
        long remain = s.Count - 1;
        foreach (var mod in s)
        {
            ModInt[] newDP = new ModInt[X + 1];

            for (int i = 0; i < dp.Length; i++)
            {
                newDP[i % mod] += dp[i];
                newDP[i] += dp[i] * remain;
            }
            remain--;
            dp = newDP;
        }
        ModInt res = 0;
        for (int i = 0; i < dp.Length; i++)
        {
            res += dp[i] * i;
        }
        Console.WriteLine(res);
    }
}

struct ModInt
{
    static readonly int MOD = 1000000007;
    long Data;
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public ModInt(long data) { if ((Data = data % MOD) < 0) Data += MOD; }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static implicit operator long(ModInt modInt) => modInt.Data;
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static implicit operator ModInt(long val) => new ModInt(val);
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static ModInt operator +(ModInt a, ModInt b)
    {
        long res = a.Data + b.Data;
        return new ModInt() { Data = res >= MOD ? res - MOD : res };
    }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static ModInt operator +(ModInt a, long b) => new ModInt(a.Data + b);
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static ModInt operator -(ModInt a, ModInt b)
    {
        long res = a.Data - b.Data;
        return new ModInt() { Data = res < 0 ? res + MOD : res };
    }
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static ModInt operator -(ModInt a, long b) => new ModInt(a.Data - b);
    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    public static ModInt operator /(ModInt a, ModInt b) => a * Power(b, MOD - 2);

    [MethodImpl(MethodImplOptions.AggressiveInlining)]
    static ModInt Power(ModInt n, long m)
    {
        ModInt res = 1;
        while (m > 0)
        {
            if ((m & 1) == 1) res *= n;
            n *= n;
            m >>= 1;
        }
        return res;
    }
}