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)

use std::io::Read;

const MAX: i64 = 100_001;
const MOD: i64 = 1_000_000_007;

fn rec(i: usize, n: usize, x: i64, s: &Vec<i64>, dp: &mut Vec<Vec<i64>>) -> i64 {
    if i == n {
        return x;
    }

    if dp[i][x as usize] < 0 {
        dp[i][x as usize] = rec(i + 1, n, x % s[i], s, dp) as i64;
        dp[i][x as usize] += (rec(i + 1, n, x, s, dp) * (n - i - 1) as i64) % MOD as i64;
        dp[i][x as usize] %= MOD;
        return dp[i][x as usize];
    }
    return dp[i][x as usize];
}

fn main() {
    let mut buf = String::new();
    std::io::stdin().read_to_string(&mut buf).unwrap();
    let mut iter = buf.split_whitespace();
    let n = iter.next().unwrap().parse().unwrap();
    let x = iter.next().unwrap().parse().unwrap();
    let mut s: Vec<i64> = (0..n)
        .map(|_| iter.next().unwrap().parse().unwrap())
        .collect();
    s.sort_by_key(|&value| -value);

    let mut dp: Vec<Vec<i64>> = vec![vec![-1; MAX as usize]; n + 1];
    println!("{}", rec(0, n, x, &s, &mut dp));
}