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 self::mod_int::ModInt;

const MOD: usize = 1e9 as usize + 7;

fn main() {
    let s = std::io::stdin();
    let mut sc = Scanner { stdin: s.lock() };
    let n: usize = sc.read();
    let x: usize = sc.read();
    let mut s: Vec<usize> = sc.vec(n);
    s.sort();
    s.reverse();
    let mut dp = vec![ModInt::new(0); x + 1];
    dp[x] = ModInt::new(1);
    for i in 0..n {
        let a = s[i];
        let p = ModInt::new(n - i).inv();
        let mut next = vec![ModInt::new(0); x + 1];

        for x in 0..(x + 1) {
            next[x % a] += dp[x] * p;
            next[x] += dp[x] * (ModInt::new(1) - p);
        }
        dp = next;
    }
    let mut fact = ModInt::new(1);
    for i in 1..(n + 1) {
        fact *= i;
    }

    let sum = dp
        .into_iter()
        .enumerate()
        .map(|(i, x)| x * i)
        .fold(ModInt::new(0), |sum, a| sum + a);
    println!("{}", (sum * fact).0);
}

pub mod mod_int {
    use super::MOD;
    use std::ops::{Add, AddAssign, Mul, MulAssign, Sub, SubAssign};

    type Num = usize;

    #[derive(Clone, Copy)]
    pub struct ModInt<T: Copy + Clone>(pub T);

    impl Add<ModInt<Num>> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn add(self, rhs: ModInt<Num>) -> ModInt<Num> {
            self + rhs.0
        }
    }

    impl Add<Num> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn add(self, rhs: Num) -> ModInt<Num> {
            let mut t = rhs + self.0;
            if t >= MOD {
                t = t - MOD;
            }
            ModInt(t)
        }
    }

    impl Sub<Num> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn sub(self, rhs: Num) -> ModInt<Num> {
            let rhs = if rhs >= MOD { rhs % MOD } else { rhs };
            let value = if self.0 < rhs { self.0 + MOD } else { self.0 };
            ModInt(value - rhs)
        }
    }

    impl Sub<ModInt<Num>> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn sub(self, rhs: ModInt<Num>) -> ModInt<Num> {
            self - rhs.0
        }
    }

    impl AddAssign<Num> for ModInt<Num> {
        fn add_assign(&mut self, other: Num) {
            *self = *self + other;
        }
    }

    impl AddAssign<ModInt<Num>> for ModInt<Num> {
        fn add_assign(&mut self, other: ModInt<Num>) {
            *self = *self + other;
        }
    }

    impl SubAssign<Num> for ModInt<Num> {
        fn sub_assign(&mut self, other: Num) {
            *self = *self - other;
        }
    }

    impl SubAssign<ModInt<Num>> for ModInt<Num> {
        fn sub_assign(&mut self, other: ModInt<Num>) {
            *self = *self - other;
        }
    }

    impl Mul<ModInt<Num>> for ModInt<Num> {
        type Output = ModInt<Num>;

        fn mul(self, rhs: ModInt<Num>) -> ModInt<Num> {
            self * rhs.0
        }
    }

    impl Mul<Num> for ModInt<Num> {
        type Output = ModInt<Num>;

        fn mul(self, rhs: Num) -> ModInt<Num> {
            let t = (self.0 * rhs) % MOD;
            ModInt(t)
        }
    }

    impl MulAssign<Num> for ModInt<Num> {
        fn mul_assign(&mut self, rhs: Num) {
            *self = *self * rhs;
        }
    }

    impl MulAssign<ModInt<Num>> for ModInt<Num> {
        fn mul_assign(&mut self, rhs: ModInt<Num>) {
            *self = *self * rhs;
        }
    }

    impl ModInt<Num> {
        pub fn new(value: Num) -> Self {
            ModInt(value)
        }

        pub fn pow(self, e: usize) -> ModInt<Num> {
            let mut result = ModInt::new(1);
            let mut cur = self;
            let mut e = e;
            while e > 0 {
                if e & 1 == 1 {
                    result *= cur;
                }
                e >>= 1;
                cur *= cur;
            }
            result
        }

        pub fn inv(self) -> ModInt<usize> {
            self.pow(MOD - 2)
        }
    }
}

pub struct Scanner<R> {
    stdin: R,
}

impl<R: std::io::Read> Scanner<R> {
    pub fn read<T: std::str::FromStr>(&mut self) -> T {
        use std::io::Read;
        let buf = self
            .stdin
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b' ' || b == b'\n')
            .take_while(|&b| b != b' ' && b != b'\n')
            .collect::<Vec<_>>();
        unsafe { std::str::from_utf8_unchecked(&buf) }
            .parse()
            .ok()
            .expect("Parse error.")
    }
    pub fn vec<T: std::str::FromStr>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| self.read()).collect()
    }
    pub fn chars(&mut self) -> Vec<char> {
        self.read::<String>().chars().collect()
    }
}