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 Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.util.ArrayList;
import java.util.HashMap;
import java.util.List;
import java.util.Map;
import java.util.Scanner;

public class Main {
  public static void main(String[] args) {
    Scanner scanner = new Scanner(System.in);

    long M = 1_000_000_007;
    int N = scanner.nextInt();
    int[] A = new int[N];
    for (int i = 0; i < N; i++) A[i] = scanner.nextInt();
    int[] primes = primes(1_000_001);
    Map<Integer, Integer> lcm = new HashMap<>();
    for (int i = 0; i < N; i++) {
      Map<Integer, Integer> factors = primeFactors(A[i], primes);
      for (Map.Entry<Integer, Integer> e : factors.entrySet()) {
        lcm.put(e.getKey(), Math.max(e.getValue(), lcm.getOrDefault(e.getKey(), 0)));
      }
    }

    long AB = 1;
    for (Map.Entry<Integer, Integer> e : lcm.entrySet()) {
      AB *= modpow(e.getKey(), e.getValue(), M);
      AB %= M;
    }
    long sum = 0;
    for (int i = 0; i < N; i++) {
      sum += AB * inverse(A[i], M) % M;
      if (sum >= M) sum -= M;
    }
    System.out.println(sum);
  }

  private static long inverse(long a, long mod) {
    return modpow(a, mod - 2, mod);
  }

  private static long modpow(long a, long n, long mod) {
    long res = 1;
    while (n > 0) {
      if (n % 2 == 1) res = (res * a) % mod;
      a = (a * a) % mod;
      n >>= 1;
    }
    return res % mod;
  }

  public static int[] primes(int n) {
    boolean[] isNotPrime = new boolean[n + 1];
    isNotPrime[0] = isNotPrime[1] = true;
    for (int i = 2; i < Math.sqrt(n); i++) {
      if (!isNotPrime[i]) {
        for (int j = 2; i * j <= n; j++) {
          isNotPrime[i * j] = true;
        }
      }
    }
    final List<Integer> ls = new ArrayList<>();
    for (int i = 2; i <= n; i++) if (!isNotPrime[i]) ls.add(i);
    int[] res = new int[ls.size()];
    for (int i = 0; i < ls.size(); i++) res[i] = ls.get(i);
    return res;
  }

  public static Map<Integer, Integer> primeFactors(int n, int[] primes) {
    Map<Integer, Integer> map = new HashMap<>();
    for (int p : primes) {
      if (p * p > n) break;
      while (n % p == 0) {
        map.put(p, map.getOrDefault(p, 0) + 1);
        n /= p;
      }
    }
    if (n > 1) map.put(n, 1);
    return map;
  }
}