You are given a positive integer primeFactors. You are asked to construct a positive integer n that satisfies the following conditions:
- The number of prime factors of
n(not necessarily distinct) is at mostprimeFactors. - The number of nice divisors of
nis maximized. Note that a divisor ofnis nice if it is divisible by every prime factor ofn. For example, ifn = 12, then its prime factors are[2,2,3], then6and12are nice divisors, while3and4are 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 <= 109
Solution: Math
Time complexity: O(logn)
Space complexity: O(1)
C++
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// Author: Huahua class Solution { public: int maxNiceDivisors(int n) { constexpr int kMod = 1e9 + 7; auto powm = [](long base, int exp) { long ans = 1; while (exp) { if (exp & 1) ans = (ans * base) % kMod; base = (base * base) % kMod; exp >>= 1; } return ans; }; if (n <= 3) return n; switch (n % 3) { case 0: return powm(3, n / 3); case 1: return (powm(3, n / 3 - 1) * 4) % kMod; case 2: return (powm(3, n / 3) * 2) % kMod; } return -1; } }; |