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)]
use std::char::from_digit;
#[allow(unused_imports)]
use std::cmp::{max, min};
#[allow(unused_imports)]
use std::collections::{BTreeMap, BTreeSet, BinaryHeap, HashMap, HashSet, VecDeque};
#[allow(unused_imports)]
use std::fs::File;
#[allow(unused_imports)]
use std::io::prelude::*;
#[allow(unused_imports)]
use std::io::{stdin, stdout, BufReader, BufWriter, Stdin, Stdout, Write};
#[allow(unused_imports)]
use std::mem::swap;
#[allow(unused_imports)]
use std::str::FromStr;
 
type BoxResult<T> = Result<T, Box<std::error::Error>>;
 
const N: usize = 1_000_000;
const MOD: u64 = 1_000_000_000 + 7;
 
fn main() {
    let mut scanner = Scanner::new(stdin());
    let _n = scanner.read::<usize>().unwrap();
    let arr = scanner.read_vector::<u64>().unwrap();
    let ans = solve(&arr);
    println!("{}", ans);
}
 
fn solve(arr: &[u64]) -> u64 {
    let primes = eratosthenes(N).into_iter().collect::<HashSet<_>>();
    let mut prime_counts = HashMap::new();
    for val in arr {
        let counts = prime_factorize(*val, &primes);
        for (p, c) in counts.into_iter() {
            if prime_counts.contains_key(&p) {
                let count = prime_counts.get_mut(&p).unwrap();
                *count = max(c, *count);
            } else {
                prime_counts.insert(p, c);
            }
        }
    }
    let all = prime_counts
        .into_iter()
        .fold(1, |acc, (p, c)| (acc * p.pow(c)) % MOD);
    let mut sum = 0;
    for val in arr {
        sum += all * power(*val, MOD - 2, MOD);
        sum %= MOD;
    }
    sum
}
 
fn eratosthenes(n: usize) -> Vec<u64> {
    let mut primes = vec![];
    let mut is_prime = vec![true; n + 1];
    for i in 2..(n + 1) {
        if !is_prime[i] {
            continue;
        }
        primes.push(i as u64);
        for j in 1..(n / i + 1) {
            is_prime[i * j] = false;
        }
    }
    primes
}
 
fn prime_factorize(n: u64, primes: &HashSet<u64>) -> Vec<(u64, u32)> {
    let mut factors = vec![];
    let mut val = 1;
    while val * val <= n {
        if n % val == 0 {
            if primes.contains(&val) {
                let count = calc_log(n, val);
                factors.push((val, count));
            }
            let d = n / val;
            if d != val && primes.contains(&d) {
                let count = calc_log(n, d);
                factors.push((d, count));
            }
        }
        val += 1;
    }
    factors
}
 
fn calc_log(mut n: u64, m: u64) -> u32 {
    let mut count = 0;
    while n % m == 0 {
        count += 1;
        n /= m;
    }
    count
}
 
fn power(x: u64, mut k: u64, modulo: u64) -> u64 {
    let mut pows = vec![x];
    for i in 0..63 {
        let square = (pows[i] * pows[i]) % modulo;
        pows.push(square);
    }
    let mut pow = 1;
    for i in 0..64 {
        let flag = k & 1;
        k >>= 1;
        if flag == 1 {
            pow = (pow * pows[i]) % modulo;
        }
    }
    pow
}
 
#[allow(dead_code)]
struct Scanner<R> {
    reader: BufReader<R>,
    buffer: String,
}
impl<R> Scanner<R>
where
    R: Read,
{
    #[allow(dead_code)]
    fn new(inner: R) -> Scanner<R> {
        let reader = BufReader::new(inner);
        let buffer = String::new();
        Scanner {
            reader: reader,
            buffer: buffer,
        }
    }
    #[allow(dead_code)]
    fn with_capacity(capacity: usize, inner: R) -> Scanner<R> {
        let reader = BufReader::with_capacity(capacity, inner);
        let buffer = String::new();
        Scanner {
            reader: reader,
            buffer: buffer,
        }
    }
    #[allow(dead_code)]
    fn read_line(&mut self) -> BoxResult<String> {
        self.reader.read_line(&mut self.buffer)?;
        let cloned = self.buffer.trim().to_string();
        self.buffer.clear();
        Ok(cloned)
    }
    #[allow(dead_code)]
    fn read<T>(&mut self) -> BoxResult<T>
    where
        T: FromStr,
        T::Err: std::fmt::Debug,
        T::Err: std::error::Error,
        T::Err: 'static,
    {
        self.reader.read_line(&mut self.buffer)?;
        let parsed = self.buffer.trim().parse()?;
        self.buffer.clear();
        Ok(parsed)
    }
    #[allow(dead_code)]
    fn read_two<T, U>(&mut self) -> BoxResult<(T, U)>
    where
        T: FromStr,
        T: Copy,
        T::Err: std::fmt::Debug,
        T::Err: std::error::Error,
        T::Err: 'static,
        U: FromStr,
        U: Copy,
        U::Err: std::fmt::Debug,
        U::Err: std::error::Error,
        U::Err: 'static,
    {
        self.reader.read_line(&mut self.buffer)?;
        let (a, b) = {
            let params = self.buffer.trim().split_whitespace().collect::<Vec<_>>();
            let a = params[0].parse()?;
            let b = params[1].parse()?;
            (a, b)
        };
        self.buffer.clear();
        Ok((a, b))
    }
    #[allow(dead_code)]
    fn read_three<T>(&mut self) -> BoxResult<(T, T, T)>
    where
        T: FromStr,
        T: Copy,
        T::Err: std::fmt::Debug,
        T::Err: std::error::Error,
        T::Err: 'static,
    {
        let nums = self.read_vector()?;
        Ok((nums[0], nums[1], nums[2]))
    }
    #[allow(dead_code)]
    fn read_vector<T>(&mut self) -> BoxResult<Vec<T>>
    where
        T: FromStr,
        T::Err: std::fmt::Debug,
        T::Err: std::error::Error,
        T::Err: 'static,
    {
        self.reader.read_line(&mut self.buffer)?;
        let parsed_vec = self
            .buffer
            .trim()
            .split_whitespace()
            .map(|s| s.parse())
            .collect::<Result<Vec<_>, _>>()?;
        self.buffer.clear();
        Ok(parsed_vec)
    }
    #[allow(dead_code)]
    fn read_matrix<T>(&mut self, lines: usize) -> BoxResult<Vec<Vec<T>>>
    where
        T: FromStr,
        T::Err: std::fmt::Debug,
        T::Err: std::error::Error,
        T::Err: 'static,
    {
        let mut parsed_matrix = vec![];
        for _ in 0..lines {
            self.reader.read_line(&mut self.buffer)?;
            parsed_matrix.push(
                self.buffer
                    .trim()
                    .split_whitespace()
                    .map(|s| s.parse())
                    .collect::<Result<Vec<_>, _>>()?,
            );
            self.buffer.clear();
        }
        Ok(parsed_matrix)
    }
    #[allow(dead_code)]
    fn read_chars(&mut self) -> BoxResult<Vec<char>> {
        self.reader.read_line(&mut self.buffer)?;
        let chars = self.buffer.trim().chars().collect::<Vec<_>>();
        self.buffer.clear();
        Ok(chars)
    }
}