You are given a positive integer primeFactors. You are asked to construct a positive integer n that satisfies the following conditions:
n (not necessarily distinct) is at most primeFactors.n is maximized. Note that a divisor of n is nice if it is divisible by every prime factor of n. For example, if n = 12, then its prime factors are [2,2,3], then 6 and 12 are nice divisors, while 3 and 4 are not.Return the number of nice divisors of n. Since that number can be too large, return it modulo 109 + 7.
Note that a prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. The prime factors of a number n is a list of prime numbers such that their product equals n.
Example 1:
Input: primeFactors = 5 Output: 6 Explanation: 200 is a valid value of n. It has 5 prime factors: [2,2,2,5,5], and it has 6 nice divisors: [10,20,40,50,100,200]. There is not other value of n that has at most 5 prime factors and more nice divisors.
Example 2:
Input: primeFactors = 8 Output: 18
Constraints:
1 <= primeFactors <= 109The task requires maximizing the number of nice divisors of n. To achieve this, we need to distribute the prime factors efficiently. When prime factors are distributed as powers, the number of divisors is maximized. The idea is similar to maximizing the product by dividing a number into parts such that the multiplication of parts is maximized, leveraging the properties of exponentiation.
Key Idea: Divide the factors into parts 3 and 2, as the number 3 contributes more efficiently to the product than smaller numbers, except when we have less than 3 factors where we prioritize 2.
This implementation optimizes the formation of n by maximizing the use of the factors. The use of pow optimally handles exponentiation with modulo operation effectively for large arguments due to the constraint size.
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Time Complexity: O(log n) — due to the use of pow which uses exponentiation by squaring.
Space Complexity: O(1) — only a constant amount of additional space is used.
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