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 Rust (1.15.1) (judge verdict: Accepted)

fn read<T: std::str::FromStr>() -> T {
    let mut s = String::new();
    std::io::stdin().read_line(&mut s).ok();
    s.trim().parse().ok().unwrap()
}

fn read_vec<T: std::str::FromStr>() -> Vec<T> {
    read::<String>()
        .split_whitespace()
        .map(|e| e.parse().ok().unwrap())
        .collect()
}

fn main() {
    let v: Vec<usize> = read_vec();
    let (n, x) = (v[0], v[1]);
    let mut s = read_vec::<usize>();
    s.sort();
    s.reverse();

    let big_int: u64 = 1000_000_000 + 7;
    let mut dp = vec![0u64; x + 1];
    dp[x] = 1;
    for si in 0..s.len() {
        let snum = s[si];
        let mut new_dp = vec![0; x + 1];
        for xnum in 0..x + 1 {
            new_dp[xnum] += dp[xnum] * (n - si - 1) as u64;
            new_dp[xnum] %= big_int;

            new_dp[xnum % snum] += dp[xnum];
            new_dp[xnum % snum] %= big_int;
        }
        dp = new_dp;
    }
    let mut ans = 0u64;
    for i in 0..dp.len() {
        ans += i as u64 * dp[i];
        ans %= big_int;
    }
    println!("{0}", ans);
}