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: Wrong Answer)

pub trait Readable {
    type Output;
    fn words_count() -> usize;
    fn read_words(words: &[&str]) -> Result<Self::Output, String>;
}
#[macro_export]
macro_rules! readable {
    ( $ t : ty , |$ words : ident | $ read_words : expr ) => {
        readable!($t, 1, |$words| $read_words);
    };
    ( $ t : ty , $ words_count : expr , |$ words : ident | $ read_words : expr ) => {
        impl Readable for $t {
            type Output = $t;
            fn words_count() -> usize {
                $words_count
            }
            fn read_words($words: &[&str]) -> Result<$t, String> {
                Ok($read_words)
            }
        }
    };
}
readable!((), |_ss| ());
readable!(String, |ss| ss[0].to_string());
macro_rules ! impl_readable_for_ints { ( $ ( $ t : ty ) * ) => { $ ( impl Readable for $ t { type Output = Self ; fn words_count ( ) -> usize { 1 } fn read_words ( words : & [ & str ] ) -> Result <$ t , String > { use std :: str :: FromStr ; <$ t >:: from_str ( words [ 0 ] ) . map_err ( | _ | { format ! ( "cannot parse \"{}\" as {}" , words [ 0 ] , stringify ! ( $ t ) ) } ) } } ) * } ; }
impl_readable_for_ints ! ( i8 u8 i16 u16 i32 u32 i64 u64 isize usize f32 f64 ) ;
macro_rules ! define_one_origin_int_types { ( $ new_t : ident $ int_t : ty ) => { # [ doc = " Converts 1-origin integer into 0-origin when read from stdin." ] # [ doc = "" ] # [ doc = " # Example" ] # [ doc = "" ] # [ doc = " ```no_run" ] # [ doc = " # #[macro_use] extern crate atcoder_snippets;" ] # [ doc = " # use atcoder_snippets::read::*;" ] # [ doc = " // Stdin: \"1\"" ] # [ doc = " read!(a = usize_);" ] # [ doc = " assert_eq!(a, 0);" ] # [ doc = " ```" ] # [ allow ( non_camel_case_types ) ] pub struct $ new_t ( ) ; impl Readable for $ new_t { type Output = $ int_t ; fn words_count ( ) -> usize { 1 } fn read_words ( words : & [ & str ] ) -> Result < Self :: Output , String > { <$ int_t >:: read_words ( words ) . map ( | n | n - 1 ) } } } ; ( $ new_t : ident $ int_t : ty ; $ ( $ inner_new_t : ident $ inner_int_t : ty ) ;* ) => { define_one_origin_int_types ! ( $ new_t $ int_t ) ; define_one_origin_int_types ! ( $ ( $ inner_new_t $ inner_int_t ) ;* ) ; } ; }
define_one_origin_int_types ! ( u8_ u8 ; u16_ u16 ; u32_ u32 ; u64_ u64 ; usize_ usize ) ;
macro_rules ! impl_readable_for_tuples { ( $ t : ident $ var : ident ) => ( ) ; ( $ t : ident $ var : ident ; $ ( $ inner_t : ident $ inner_var : ident ) ;* ) => { impl_readable_for_tuples ! ( $ ( $ inner_t $ inner_var ) ;* ) ; impl <$ t : Readable , $ ( $ inner_t : Readable ) ,*> Readable for ( $ t , $ ( $ inner_t ) ,* ) { type Output = ( <$ t >:: Output , $ ( <$ inner_t >:: Output ) ,* ) ; fn words_count ( ) -> usize { let mut n = <$ t >:: words_count ( ) ; $ ( n += <$ inner_t >:: words_count ( ) ; ) * n } # [ allow ( unused_assignments ) ] fn read_words ( words : & [ & str ] ) -> Result < Self :: Output , String > { let mut start = 0 ; let $ var = <$ t >:: read_words ( & words [ start .. start +<$ t >:: words_count ( ) ] ) ?; start += <$ t >:: words_count ( ) ; $ ( let $ inner_var = <$ inner_t >:: read_words ( & words [ start .. start +<$ inner_t >:: words_count ( ) ] ) ?; start += <$ inner_t >:: words_count ( ) ; ) * Ok ( ( $ var , $ ( $ inner_var ) ,* ) ) } } } ; }
impl_readable_for_tuples ! ( T4 x4 ; T3 x3 ; T2 x2 ; T1 x1 ) ;
pub trait ReadableFromLine {
    type Output;
    fn read_line(line: &str) -> Result<Self::Output, String>;
}
fn split_into_words(line: &str) -> Vec<&str> {
    #[allow(deprecated)]
    line.trim_right_matches('\n').split_whitespace().collect()
}
impl<T: Readable> ReadableFromLine for T {
    type Output = T::Output;
    fn read_line(line: &str) -> Result<T::Output, String> {
        let words = split_into_words(line);
        if words.len() != T::words_count() {
            return Err(format!(
                "line \"{}\" has {} words, expected {}",
                line,
                words.len(),
                T::words_count()
            ));
        }
        T::read_words(&words)
    }
}
impl<T: Readable> ReadableFromLine for Vec<T> {
    type Output = Vec<T::Output>;
    fn read_line(line: &str) -> Result<Self::Output, String> {
        let n = T::words_count();
        let words = split_into_words(line);
        if words.len() % n != 0 {
            return Err(format!(
                "line \"{}\" has {} words, expected multiple of {}",
                line,
                words.len(),
                n
            ));
        }
        let mut result = Vec::new();
        for chunk in words.chunks(n) {
            match T::read_words(chunk) {
                Ok(v) => result.push(v),
                Err(msg) => {
                    let flagment_msg = if n == 1 {
                        format!("word {}", result.len())
                    } else {
                        let l = result.len();
                        format!("words {}-{}", n * l + 1, (n + 1) * l)
                    };
                    return Err(format!("{} of line \"{}\": {}", flagment_msg, line, msg));
                }
            }
        }
        Ok(result)
    }
}
impl<T: Readable, U: Readable> ReadableFromLine for (T, Vec<U>) {
    type Output = (T::Output, <Vec<U> as ReadableFromLine>::Output);
    fn read_line(line: &str) -> Result<Self::Output, String> {
        let n = T::words_count();
        #[allow(deprecated)]
        let trimmed = line.trim_right_matches('\n');
        let words_and_rest: Vec<&str> = trimmed.splitn(n + 1, ' ').collect();
        if words_and_rest.len() < n {
            return Err(format!(
                "line \"{}\" has {} words, expected at least {}",
                line,
                words_and_rest.len(),
                n
            ));
        }
        let words = &words_and_rest[..n];
        let empty_str = "";
        let rest = words_and_rest.get(n).unwrap_or(&empty_str);
        Ok((T::read_words(words)?, Vec::<U>::read_line(rest)?))
    }
}
macro_rules ! impl_readable_from_line_for_tuples_with_vec { ( $ t : ident $ var : ident ) => ( ) ; ( $ t : ident $ var : ident ; $ ( $ inner_t : ident $ inner_var : ident ) ;+ ) => { impl_readable_from_line_for_tuples_with_vec ! ( $ ( $ inner_t $ inner_var ) ;+ ) ; impl <$ t : Readable , $ ( $ inner_t : Readable ) ,+ , U : Readable > ReadableFromLine for ( $ t , $ ( $ inner_t ) ,+ , Vec < U > ) { type Output = ( $ t :: Output , $ ( $ inner_t :: Output ) ,+ , Vec < U :: Output > ) ; fn read_line ( line : & str ) -> Result < Self :: Output , String > { let mut n = $ t :: words_count ( ) ; $ ( n += $ inner_t :: words_count ( ) ; ) + # [ allow ( deprecated ) ] let trimmed = line . trim_right_matches ( '\n' ) ; let words_and_rest : Vec <& str > = trimmed . splitn ( n + 1 , ' ' ) . collect ( ) ; if words_and_rest . len ( ) < n { return Err ( format ! ( "line \"{}\" has {} words, expected at least {}" , line , words_and_rest . len ( ) , n ) ) ; } let words = & words_and_rest [ .. n ] ; let empty_str = "" ; let rest = words_and_rest . get ( n ) . unwrap_or ( & empty_str ) ; let ( $ var , $ ( $ inner_var ) ,* ) = < ( $ t , $ ( $ inner_t ) ,+ ) >:: read_words ( words ) ?; Ok ( ( $ var , $ ( $ inner_var ) ,* , Vec ::< U >:: read_line ( rest ) ? ) ) } } } ; }
impl_readable_from_line_for_tuples_with_vec ! ( T4 t4 ; T3 t3 ; T2 t2 ; T1 t1 ) ;
pub fn read<T: ReadableFromLine>() -> T::Output {
    let mut line = String::new();
    std::io::stdin().read_line(&mut line).unwrap();
    T::read_line(&line).unwrap()
}
#[macro_export]
macro_rules ! read { ( ) => { let mut line = String :: new ( ) ; std :: io :: stdin ( ) . read_line ( & mut line ) . unwrap ( ) ; } ; ( $ pat : pat = $ t : ty ) => { let $ pat = read ::<$ t > ( ) ; } ; ( $ ( $ pat : pat = $ t : ty ) ,+ ) => { read ! ( ( $ ( $ pat ) ,* ) = ( $ ( $ t ) ,* ) ) ; } ; }
#[macro_export]
macro_rules ! readls { ( $ ( $ pat : pat = $ t : ty ) ,+ ) => { $ ( read ! ( $ pat = $ t ) ; ) * } ; }
pub fn readx<T: ReadableFromLine>() -> Vec<T::Output> {
    use std::io::{self, BufRead};
    let stdin = io::stdin();
    let result = stdin
        .lock()
        .lines()
        .map(|line_result| {
            let line = line_result.expect("read from stdin failed");
            T::read_line(&line).unwrap()
        })
        .collect();
    result
}
#[macro_export]
macro_rules ! readx_loop { ( |$ pat : pat = $ t : ty | $ body : expr ) => { use std :: io :: BufRead ; let stdin = std :: io :: stdin ( ) ; for line in stdin . lock ( ) . lines ( ) { let line = line . expect ( "read from stdin failed" ) ; let $ pat = <$ t >:: read_line ( & line ) . unwrap ( ) ; $ body } } ; ( |$ ( $ pat : pat = $ t : ty ) ,*| $ body : expr ) => { readx_loop ! ( | ( $ ( $ pat ) ,* ) = ( $ ( $ t ) ,* ) | $ body ) ; } ; }
pub fn readn<T: ReadableFromLine>(n: usize) -> Vec<T::Output> {
    use std::io::{self, BufRead};
    let stdin = io::stdin();
    let result: Vec<T::Output> = stdin
        .lock()
        .lines()
        .take(n)
        .map(|line_result| {
            let line = line_result.expect("read from stdin failed");
            T::read_line(&line).unwrap()
        })
        .collect();
    if result.len() < n {
        panic!(
            "expected reading {} lines, but only {} lines are read",
            n,
            result.len()
        );
    }
    result
}
#[macro_export]
macro_rules ! readn_loop { ( $ n : expr , |$ pat : pat = $ t : ty | $ body : expr ) => { use std :: io :: BufRead ; let stdin = std :: io :: stdin ( ) ; { let mut lock = stdin . lock ( ) ; for _ in 0 ..$ n { let mut line = String :: new ( ) ; lock . read_line ( & mut line ) . expect ( "read from stdin failed" ) ; let $ pat = <$ t >:: read_line ( & line ) . unwrap ( ) ; $ body } } } ; ( $ n : expr , |$ ( $ pat : pat = $ t : ty ) ,*| $ body : expr ) => { readn_loop ! ( $ n , | ( $ ( $ pat ) ,* ) = ( $ ( $ t ) ,* ) | $ body ) ; } ; }
pub trait Words {
    fn read<T: Readable>(&self) -> T::Output;
}
impl<'a> Words for [&'a str] {
    fn read<T: Readable>(&self) -> T::Output {
        T::read_words(self).unwrap()
    }
}
impl<'a> Words for &'a str {
    fn read<T: Readable>(&self) -> T::Output {
        T::read_words(&[self]).unwrap()
    }
}

#[derive(Clone)]
pub struct StepBy<I> {
    iter: I,
    step: usize,
    first_take: bool,
}
impl<I: Iterator> Iterator for StepBy<I> {
    type Item = I::Item;
    fn next(&mut self) -> Option<Self::Item> {
        if self.first_take {
            self.first_take = false;
            self.iter.next()
        } else {
            self.iter.nth(self.step)
        }
    }
}
#[derive(Clone)]
pub struct LScan<I: Iterator, S: Clone, F: FnMut(&S, I::Item) -> S> {
    iter: I,
    state: Option<S>,
    f: F,
}
impl<I: Iterator, S: Clone, F> Iterator for LScan<I, S, F>
where
    F: FnMut(&S, I::Item) -> S,
{
    type Item = S;
    fn next(&mut self) -> Option<S> {
        if self.state.is_none() {
            return None;
        }
        let state_inner = self.state.take().unwrap();
        if let Some(item) = self.iter.next() {
            self.state = Some((self.f)(&state_inner, item));
        }
        Some(state_inner)
    }
}
pub struct Flatten<I: Iterator>
where
    I::Item: IntoIterator,
{
    outer_iter: I,
    inner_iter: Option<<<I as Iterator>::Item as IntoIterator>::IntoIter>,
}
impl<I, J> Iterator for Flatten<I>
where
    I: Iterator<Item = J>,
    J: IntoIterator,
{
    type Item = <<J as IntoIterator>::IntoIter as Iterator>::Item;
    fn next(&mut self) -> Option<J::Item> {
        loop {
            if let Some(inner_iter) = self.inner_iter.as_mut() {
                if let item @ Some(_) = inner_iter.next() {
                    return item;
                }
            }
            match self.outer_iter.next() {
                None => return None,
                Some(inner) => self.inner_iter = Some(inner.into_iter()),
            }
        }
    }
}
pub struct GroupBy<K: Eq, I: Iterator, F: FnMut(&I::Item) -> K> {
    cur: Option<(I::Item, K)>,
    iter: I,
    key_fn: F,
}
impl<K: Eq, I: Iterator, F: FnMut(&I::Item) -> K> Iterator for GroupBy<K, I, F> {
    type Item = Vec<I::Item>;
    fn next(&mut self) -> Option<Vec<I::Item>> {
        let cur = self.cur.take();
        cur.map(|(item, key)| {
            let mut group = vec![item];
            loop {
                let next = self.iter.next();
                match next {
                    Some(next_item) => {
                        let next_key = (self.key_fn)(&next_item);
                        if key == next_key {
                            group.push(next_item);
                        } else {
                            self.cur = Some((next_item, next_key));
                            break;
                        }
                    }
                    None => {
                        self.cur = None;
                        break;
                    }
                }
            }
            group
        })
    }
}
pub trait IteratorExt: Iterator {
    fn step_by_(self, step: usize) -> StepBy<Self>
    where
        Self: Sized,
    {
        assert_ne!(step, 0);
        StepBy {
            iter: self,
            step: step - 1,
            first_take: true,
        }
    }
    fn for_each<F: FnMut(Self::Item)>(self, mut f: F)
    where
        Self: Sized,
    {
        for item in self {
            f(item);
        }
    }
    fn lscan<S: Clone, F>(self, state: S, f: F) -> LScan<Self, S, F>
    where
        Self: Sized,
        F: FnMut(&S, Self::Item) -> S,
    {
        LScan {
            iter: self,
            state: Some(state),
            f: f,
        }
    }
    fn get_unique(mut self) -> Option<Self::Item>
    where
        Self: Sized,
        Self::Item: Eq,
    {
        let first_opt = self.next();
        first_opt.and_then(|first| {
            if self.all(|item| item == first) {
                Some(first)
            } else {
                None
            }
        })
    }
    fn flatten(mut self) -> Flatten<Self>
    where
        Self: Sized,
        Self::Item: IntoIterator,
    {
        let inner_opt = self.next();
        Flatten {
            outer_iter: self,
            inner_iter: inner_opt.map(|inner| inner.into_iter()),
        }
    }
    fn group_by<K: Eq, F: FnMut(&Self::Item) -> K>(mut self, mut f: F) -> GroupBy<K, Self, F>
    where
        Self: Sized,
    {
        let next = self.next();
        GroupBy {
            cur: next.map(|item| {
                let key = f(&item);
                (item, key)
            }),
            iter: self,
            key_fn: f,
        }
    }
    fn join(mut self, sep: &str) -> String
    where
        Self: Sized,
        Self::Item: std::fmt::Display,
    {
        let mut result = String::new();
        if let Some(first) = self.next() {
            result.push_str(&format!("{}", first));
        }
        for s in self {
            result.push_str(&format!("{}{}", sep, s));
        }
        result
    }
    fn cat(self) -> String
    where
        Self: Sized,
        Self::Item: std::fmt::Display,
    {
        self.join("")
    }
}
impl<I: Iterator> IteratorExt for I {}
pub struct Unfold<T, F>
where
    F: FnMut(&T) -> Option<T>,
{
    state: Option<T>,
    f: F,
}
impl<T, F> Iterator for Unfold<T, F>
where
    F: FnMut(&T) -> Option<T>,
{
    type Item = T;
    fn next(&mut self) -> Option<T> {
        if self.state.is_none() {
            return None;
        }
        let state_inner = self.state.take().unwrap();
        self.state = (self.f)(&state_inner);
        Some(state_inner)
    }
}
pub fn unfold<T, F>(init: T, f: F) -> Unfold<T, F>
where
    F: FnMut(&T) -> Option<T>,
{
    Unfold {
        state: Some(init),
        f: f,
    }
}
pub struct Iterate<T, F>
where
    F: FnMut(&T) -> T,
{
    state: T,
    f: F,
}
impl<T, F> Iterator for Iterate<T, F>
where
    F: FnMut(&T) -> T,
{
    type Item = T;
    fn next(&mut self) -> Option<T> {
        use std::mem::swap;
        let mut state = (self.f)(&self.state);
        swap(&mut state, &mut self.state);
        Some(state)
    }
}
pub fn iterate<T, F>(init: T, f: F) -> Iterate<T, F>
where
    F: FnMut(&T) -> T,
{
    Iterate { state: init, f: f }
}

use std::collections::HashSet;

const MOD: u64 = 1_000_000_007;

#[macro_export]
macro_rules! dbg {
    ( $ e : expr ) => {{
        use std::io::{self, Write};
        let result = $e;
        writeln!(
            io::stderr(),
            "{}: {} = {:?}",
            line!(),
            stringify!($e),
            result
        )
            .unwrap();
        result
    }};
}

fn calc(x: u32, ss: &mut HashSet<u32>, facts: &[u64]) -> u64 {
    if ss.is_empty() {
        x as u64
    } else {
        let mut lesser = HashSet::new();
        let mut greater = HashSet::new();
        for s in ss.iter().cloned() {
            if s < x {
                lesser.insert(s);
            } else {
                greater.insert(s);
            }
        }

        let mut ans = 0;
        if greater.len() >= 1 && !greater.contains(&x) {
            ans += calc(x, &mut lesser, facts) * facts[greater.len()] % MOD;
        }
        for l in lesser.iter().cloned() {
            ss.remove(&l);
            ans += calc(x % l, ss, facts);
            ss.insert(l);
        }

        ans % MOD
    }
}

fn main() {
    read!(_ = (), x = u32);
    read!(ss = Vec<u32>);

    let facts = (1..(ss.len() as u64 + 1))
        .lscan(1, |&acc, next| acc * next % MOD)
        .collect::<Vec<_>>();

    dbg!(&facts);

    if ss.contains(&1) {
        println!("0");
    } else {
        println!("{}", calc(x, &mut ss.into_iter().collect(), &facts));
    }
}