Problem:
Score : 500 points

Problem Statement

Given are N positive integers A_1,...,A_N.

Consider positive integers B_1, ..., B_N that satisfy the following condition.

Condition: For any i, j such that 1 \leq i < j \leq N, A_i B_i = A_j B_j holds.

Find the minimum possible value of B_1 + ... + B_N for such B_1,...,B_N.

Since the answer can be enormous, print the sum modulo (10^9 +7).

Constraints

1 \leq N \leq 10^4

1 \leq A_i \leq 10^6

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
A_1 ... A_N

Output

Print the minimum possible value of B_1 + ... + B_N for B_1,...,B_N that satisfy the condition, modulo (10^9 +7).

Sample Input 1

3
2 3 4

Sample Output 1

13

Let B_1=6, B_2=4, and B_3=3, and the condition will be satisfied.

Sample Input 2

5
12 12 12 12 12

Sample Output 2

5

We can let all B_i be 1.

Sample Input 3

3
1000000 999999 999998

Sample Output 3

996989508

Print the sum modulo (10^9+7).

Solution in Rust (1.15.1) (judge verdict: Accepted)

#![allow(unused_imports, unused_macros, dead_code)]

macro_rules! min {
    (.. $x:expr) => {{
        let mut it = $x.iter();
        it.next().map(|z| it.fold(z, |x, y| min!(x, y)))
    }};
    ($x:expr) => ($x);
    ($x:expr, $($ys:expr),*) => {{
        let t = min!($($ys),*);
        if $x < t { $x } else { t }
    }}
}
macro_rules! max {
    (.. $x:expr) => {{
        let mut it = $x.iter();
        it.next().map(|z| it.fold(z, |x, y| max!(x, y)))
    }};
    ($x:expr) => ($x);
    ($x:expr, $($ys:expr),*) => {{
        let t = max!($($ys),*);
        if $x > t { $x } else { t }
    }}
}
macro_rules! ewriteln {
    ($($args:expr),*) => { let _ = writeln!(&mut std::io::stderr(), $($args),*); };
}
macro_rules! trace {
    ($x:expr) => { ewriteln!(">>> {} = {:?}", stringify!($x), $x) };
    ($($xs:expr),*) => { trace!(($($xs),*)) }
}
macro_rules! put {
    (.. $x:expr) => {{
        let mut it = $x.iter();
        if let Some(x) = it.next() { print!("{}", x); }
        for x in it { print!(" {}", x); }
        println!("");
    }};
    ($x:expr) => { println!("{}", $x) };
    ($x:expr, $($xs:expr),*) => { print!("{} ", $x); put!($($xs),*) }
}

const M: i64 = 1_000_000_007;

fn prime_factorize(n: u64) -> DefaultMap<u64, usize> {
    let mut m = n;
    let mut r = DefaultMap::new(0);
    for x in 2..n + 1 {
        if m == 1 { break }
        if x * x > n {
            r[m] = 1;
            break
        }
        if n % x != 0 { continue }
        let mut c = 0;
        while m % x == 0 { m /= x; c += 1; }
        r[x] = c;
    }
    r
}

struct DefaultMap<S, T> {
    data: std::collections::HashMap<S, T>,
    default: T,
}
impl<S: Eq + std::hash::Hash, T> DefaultMap<S, T> {
    fn new(default: T) -> DefaultMap<S, T> {
        DefaultMap {
            data: std::collections::HashMap::new(),
            default: default
        }
    }
    fn keys(&self) -> std::collections::hash_map::Keys<S, T> { self.data.keys() }
    fn iter(&self) -> std::collections::hash_map::Iter<S, T> { self.data.iter() }
    fn len(&self) -> usize { self.data.len() }
}
impl<S: Eq + std::hash::Hash, T> std::ops::Index<S> for DefaultMap<S, T> {
    type Output = T;
    fn index(&self, key: S) -> &Self::Output {
        if let Some(val) = self.data.get(&key) {
            val
        } else {
            &self.default
        }
    }
}
impl<S: Eq + std::hash::Hash + Clone, T: Clone> std::ops::IndexMut<S> for DefaultMap<S, T> {
    fn index_mut(&mut self, key: S) -> &mut Self::Output {
        self.data.entry(key.clone()).or_insert(self.default.clone());
        self.data.get_mut(&key).unwrap()
    }
}

#[derive(Debug, PartialEq, Eq, Clone, Copy)]
struct ModInt(i64, i64);  // (residual, modulo)
impl ModInt {
    fn new(residual: i64, modulo: i64) -> ModInt {
        if residual >= modulo { ModInt(residual % modulo, modulo) }
        else if residual < 0 { ModInt((residual % modulo) + modulo, modulo) }
        else { ModInt(residual, modulo) }
    }
    fn unwrap(self) -> i64 { self.0 }
    fn inv(self) -> Self {
        fn exgcd(r0: i64, a0: i64, b0: i64, r: i64, a: i64, b: i64)
                -> (i64, i64, i64) {
            if r > 0 {
                exgcd(r, a, b, r0 % r, a0 - r0 / r * a, b0 - r0 / r * b)
            } else {
                (a0, b0, r0)
            }
        }
        let (a, _, r) = exgcd(self.0, 1, 0, self.1, 0, 1);
        if r != 1 { panic!("{:?} has no inverse!", self); }
        ModInt(((a % self.1) + self.1) % self.1, self.1)
    }
    fn pow(self, n: i64) -> Self {
        if n < 0 {
            self.pow(-n).inv()
        } else if n == 0 {
            ModInt(1, self.1)
        } else if n == 1 {
            self
        } else {
            let mut x = (self * self).pow(n / 2);
            if n % 2 == 1 { x *= self }
            x
        }
    }
}
impl std::fmt::Display for ModInt {
    fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
        write!(f, "{}", self.0)
    }
}
impl std::ops::Neg for ModInt {
    type Output = Self;
    fn neg(self) -> Self {
        if self.0 == 0 { return self }
        ModInt(self.1 - self.0, self.1)
    }
}
impl std::ops::Add<i64> for ModInt {
    type Output = Self;
    fn add(self, other: i64) -> Self { ModInt::new(self.0 + other, self.1) }
}
impl std::ops::Add for ModInt {
    type Output = Self;
    fn add(self, other: ModInt) -> Self { self + other.0 }
}
impl std::ops::Add<ModInt> for i64 {
    type Output = ModInt;
    fn add(self, other: ModInt) -> ModInt { other + self }
}
impl std::ops::AddAssign<i64> for ModInt {
    fn add_assign(&mut self, other: i64) { self.0 = ModInt::new(self.0 + other, self.1).0; }
}
impl std::ops::AddAssign for ModInt {
    fn add_assign(&mut self, other: ModInt) { *self += other.0; }
}
impl std::ops::Sub<i64> for ModInt {
    type Output = Self;
    fn sub(self, other: i64) -> Self { ModInt::new(self.0 - other, self.1) }
}
impl std::ops::Sub for ModInt {
    type Output = Self;
    fn sub(self, other: ModInt) -> Self { self - other.0 }
}
impl std::ops::Sub<ModInt> for i64 {
    type Output = ModInt;
    fn sub(self, other: ModInt) -> ModInt { ModInt::new(self - other.0, other.1) }
}
impl std::ops::SubAssign<i64> for ModInt {
    fn sub_assign(&mut self, other: i64) { self.0 = ModInt::new(self.0 - other, self.1).0; }
}
impl std::ops::SubAssign for ModInt {
    fn sub_assign(&mut self, other: ModInt) { *self -= other.0; }
}
impl std::ops::Mul<i64> for ModInt {
    type Output = Self;
    fn mul(self, other: i64) -> Self { ModInt::new(self.0 * other, self.1) }
}
impl std::ops::Mul for ModInt {
    type Output = Self;
    fn mul(self, other: ModInt) -> Self { self * other.0 }
}
impl std::ops::Mul<ModInt> for i64 {
    type Output = ModInt;
    fn mul(self, other: ModInt) -> ModInt { other * self }
}
impl std::ops::MulAssign<i64> for ModInt {
    fn mul_assign(&mut self, other: i64) { self.0 = ModInt::new(self.0 * other, self.1).0; }
}
impl std::ops::MulAssign for ModInt {
    fn mul_assign(&mut self, other: ModInt) { *self *= other.0; }
}
impl std::ops::Div for ModInt {
    type Output = Self;
    fn div(self, other: ModInt) -> Self { self * other.inv() }
}
impl std::ops::Div<i64> for ModInt {
    type Output = Self;
    fn div(self, other: i64) -> Self { self / ModInt(other, self.1) }
}
impl std::ops::Div<ModInt> for i64 {
    type Output = ModInt;
    fn div(self, other: ModInt) -> ModInt { other.inv() * self }
}
impl std::ops::DivAssign for ModInt {
    fn div_assign(&mut self, other: ModInt) { self.0 = (self.clone() / other).0; }
}
impl std::ops::DivAssign<i64> for ModInt {
    fn div_assign(&mut self, other: i64) { *self /= ModInt(other, self.1); }
}

fn main() {
    let mut sc = Scanner::new();
    let n: usize = sc.cin();
    let xs: Vec<_> = sc.vec(n);

    let mut map = DefaultMap::new(0);
    for &x in xs.iter() {
        let rx = prime_factorize(x);
        for (&p, &m) in rx.iter() {
            if map[p] < m {
                map[p] = m;
            }
        }
    }

    let mut lcm = ModInt(1, M);
    for (&p, &m) in map.iter() {
        lcm *= ModInt(p as i64, M).pow(m as i64);
    }
    // trace!(lcm);

    let mut ans = ModInt(0, M);
    for &x in xs.iter() {
        let mut r = lcm;
        r /= x as i64;
        ans += r;
    }
    put!(ans);

}

use std::io::{self, Write};
use std::str::FromStr;
use std::collections::VecDeque;

struct Scanner { stdin: io::Stdin, buffer: VecDeque<String>, }
impl Scanner {
    fn new() -> Self { Scanner { stdin: io::stdin(), buffer: VecDeque::new() } }
    fn cin<T: FromStr>(&mut self) -> T {
        while self.buffer.len() == 0 {
            let mut line = String::new();
            let _ = self.stdin.read_line(&mut line);
            for w in line.split_whitespace() {
                self.buffer.push_back(String::from(w));
            }
        }
        self.buffer.pop_front().unwrap().parse::<T>().ok().unwrap()
    }
    fn chars(&mut self) -> Vec<char> { self.cin::<String>().chars().collect() }
    fn vec<T: FromStr>(&mut self, n: usize) -> Vec<T> { (0..n).map(|_| self.cin()).collect() }
}