Leetcode #1808: Maximize Number of Nice Divisors
In this guide, we solve Leetcode #1808 Maximize Number of Nice Divisors in Python and focus on the core idea that makes the solution efficient.
You will see the intuition, the step-by-step method, and a clean Python implementation you can use in interviews.
Problem Statement
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 most primeFactors.
Quick Facts
- Difficulty: Hard
- Premium: No
- Tags: Recursion, Math, Number Theory
Intuition
There is a mathematical invariant or formula that directly leads to the result.
Using math avoids unnecessary loops and reduces complexity.
Approach
Derive the formula or update rule, then compute the answer directly.
Handle edge cases like overflow or zero carefully.
Steps:
- Identify the math relationship.
- Compute the result with a loop or formula.
- Handle edge cases.
Example
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.
Python Solution
class Solution:
def maxNiceDivisors(self, primeFactors: int) -> int:
mod = 10**9 + 7
if primeFactors < 4:
return primeFactors
if primeFactors % 3 == 0:
return pow(3, primeFactors // 3, mod) % mod
if primeFactors % 3 == 1:
return 4 * pow(3, primeFactors // 3 - 1, mod) % mod
return 2 * pow(3, primeFactors // 3, mod) % mod
Complexity
The time complexity is , and the space complexity is . The space complexity is .
Edge Cases and Pitfalls
Watch for boundary values, empty inputs, and duplicate values where applicable. If the problem involves ordering or constraints, confirm the invariant is preserved at every step.
Summary
This Python solution focuses on the essential structure of the problem and keeps the implementation interview-friendly while meeting the constraints.