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).



Sample input:
3
2 3 4


Sample output:
13

Submission status: Runtime Error

Solution code:
from collections import defaultdict
import numpy as np


def trial_division(n):
    factor = defaultdict(int)
    tmp = int(np.sqrt(n)) + 1
    for num in range(2, tmp):
        while n % num == 0:
            n //= num
            factor[num] += 1
    factor[n] += 1
    return factor


def powmod(x, y, mod):
    if y == 0:
        return 1
    return (((powmod(x, y >> 1, mod) ** 2) % mod) * x ** (y & 1)) % mod


n = int(input())
a = np.array(list(map(int, input().split())))
mod = 10**9 + 7
numbers = defaultdict(int)
for a_i in a:
    for i, v in trial_division(a_i).items():
        numbers[i] = max(v, numbers[i])

l = 1
for i, v in numbers.items():
    l *= i ** v
    l %= mod

r = powmod(a, mod-2, mod)
ans = 0
for i in r:
    ans += l * r
    ans %= mod
print(int(ans))

Submission status: Runtime Error

Solution code:
from fractions import gcd
import math
n = int(input())
a = list(map(int, input().split()))
MOD=10**9+7
lcm=1
for i in range(n):
    lcm*=a[i]//math.gcd(lcm,a[i])
res=0
for i in range(n):
    res+=lcm//a[i]
print(res%MOD)

Submission status: Runtime Error

Solution code:
import fraction
from functools import reduce
prime = 10**9 + 7
ans = 0

N = int(input())
A = list(map(int, input().split()))

def lcm_base(x, y):
    return (x * y) // fraction.gcd(x, y)

def lcm_list(numbers):
    return reduce(lcm_base, numbers, 1)

lcm = lcm_list(A)

for i in range(len(A)):
    ans += (lcm // A[i]) % prime

ans %= prime

print(ans)

Submission status: Runtime Error

Solution code:
from math import gcd

def lcm(a, b):
    return a // gcd(a, b) * b

MOD = 10**9 + 7

N = int(input())
a = [int(x) for x in input().split()]

m = 1
for e in a:
    m = lcm(m, e)

ans = 0
for e in a:
    ans += m // e
    ans %= MOD

print(ans)