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 (1.42.0) (judge verdict: Time Limit Exceeded)

use proconio::input;
use std::collections::HashMap;
use MyEither::{IsPrime, NotPrime, Undecided};

fn main() {
    input! {
        n: usize,
        a: [usize; n],
    }

    let limit = 1000000;
    let m = 1000000007;
    let mut map = HashMap::new();
    let mut sieve: Vec<MyEither<usize>> = vec![Undecided; limit + 1];
    let mut num = 2;

    while num <= limit {
        if sieve[num] == Undecided {
            sieve[num] = IsPrime;
            let mut tmp = num * num;
            while tmp <= limit {
                sieve[tmp] = match &sieve[tmp] {
                    Undecided => NotPrime(num),
                    n => *n,
                };
                tmp += num;
            }
        }
        num += 1;
    }

    for &na in &a {
        let tmp = prime_factorize_fast(na, &sieve);
        for (&key, &value) in &tmp {
            let count = map.entry(key).or_insert(0);
            if *count < value {
                *count = value;
            }
        }
    }

    let mut lcm = 1usize;
    for (key, value) in &map {
        let mut tmp = 1;
        for _ in 0..*value {
            tmp = tmp * key % m;
        }
        lcm = lcm * tmp % m;
    }

    let mut ans = 0usize;
    for &na in &a {
        ans = ans + lcm * modinv(na as isize, m as isize) as usize % m;
        ans %= m;
    }

    println!("{}", ans);
}

#[derive(Clone, Copy, PartialEq)]
enum MyEither<T> {
    IsPrime,
    NotPrime(T),
    Undecided,
}

fn prime_factorize_fast(mut n: usize, sieve: &Vec<MyEither<usize>>) -> HashMap<usize, usize> {
    let mut res = HashMap::new();

    while sieve[n] != IsPrime {
        match &sieve[n] {
            NotPrime(num) => {
                let entry = res.entry(*num).or_insert(0);
                *entry += 1;
                n = n / *num;
            }
            _ => {},
        }
    }
    let entry = res.entry(n).or_insert(0);
    *entry += 1;

    res
}

fn prime_factorize(mut n: usize) -> Vec<(usize, usize)> {
    let mut res = vec![];
    let mut a = 2;
    while a * a <= n {
        if n % a != 0 {
            a += 1;
            continue;
        }
        let mut ex = 0;

        while n % a == 0 {
            ex += 1;
            n /= a;
        }

        res.push((a, ex));
        a += 1;
    }
    if n != 1 {
        res.push((n, 1));
    }

    res
}

use std::mem;

fn modinv(mut a: isize, m: isize) -> isize {
    let mut b: isize = m;
    let mut u: isize = 1;
    let mut v = 0;

    while b != 0 {
        let t = a / b;
        a -= t * b;
        mem::swap(&mut a, &mut b);
        u -= t * v;
        mem::swap(&mut u, &mut v);
    }

    u %= m;

    if u < 0 {
        u += m;
    }
    u
}