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++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <bits/stdc++.h>

using namespace std;
using llong = long long;
using ldbl = long double;
using P = pair<llong, llong>;

#define ALL(x) x.begin(), x.end()

const llong inf = llong(1e18)+7;
const llong mod = 1e9+7;

llong N, X;
vector<llong> S;
vector<vector<llong> > memo;

llong dp(llong now, llong num){
	if(now == N)
		return num;
	if(memo[now][num] != -1)
		return memo[now][num];

	llong ret = 0;
	ret = (ret + dp(now+1, num%S[now])) % mod;
	ret = (ret + dp(now+1, num)*(N-now-1)) % mod;

	return memo[now][num] = ret;
}

int main(){

	cin >> N >> X;
	S.resize(N);
	for(auto &x : S)
		cin >> x;
	sort(ALL(S), greater<>());

	memo.resize(N, vector<llong>(X+1,-1));
	cout << dp(0,X) << endl;

	return 0;
}