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>
#define int long long
using namespace std;
const int mod = 1e9 + 7, maxN = 1e5 + 13;
int n, x, dp[maxN], dpw[maxN];
vector<int> vec;
int pw (int a, int b) {
    int ret = 1;
    for (; b; b >>= 1, a = 1ll * a * a % mod)
	if (b & 1)
	    ret = 1ll * ret * a % mod;
    return ret;
}

					
int32_t main () {
    cin >> n >> x;
    int cur = 1;
    for (int i = 0; i < n; i++) {
	int a;
	cin >> a;
	if (a <= x)
	    vec.push_back(a);
	cur = (cur * (i + 1)) % mod;
    }
    sort(vec.begin(), vec.end());

    for (int i = 0; i <= n; i++)
	dpw[i] = pw(i, mod - 2);
    
    for (int i = 0; i <= x; i++) {
	int pos = upper_bound(vec.begin(), vec.end(), i) - vec.begin();
	if (pos == 0)
	    dp[i] = i;
	for (int j = 0; j < pos; j++) {
	    //cout << i << " " << vec[j] << " " << dp[i % vec[j]] << endl;
	    dp[i] = (dp[i] + dpw[pos] * dp[i % vec[j]]) % mod;
	}
	
	//cout << i << " " << pos << " " << dp[i] << endl;
    }
    
    //cout << solve(x, n) << " " << cur  << endl;
    cout << 1ll * dp[x] * cur % mod << endl;
}