月入过万为何存不下钱？

Approach指解决问题或达成目标的具体方法或路径，包括策略、步骤和工具的选择与实施，旨在系统化、高效地实现预期结果。

To solve this problem, we need to find the maximum number of distinct prime factors for any number in the inclusive range [left, right]. The solution involves iterating through each number in the given range, computing its prime factors, and keeping track of the maximum count of distinct prime factors encountered.

1. Problem Analysis: The task is to determine the highest number of distinct prime factors for any number within a specified range. A prime factor is a prime number that divides the given number exactly.
2. Key Insight: For each number in the range [left, right], we decompose it into its distinct prime factors. The count of these distinct primes is then compared to find the maximum count across all numbers in the range.
3. Algorithm Selection:
   • Iterate over each number from left to right.
   • For each number, compute its distinct prime factors by checking divisibility starting from 2 up to the square root of the number.
   • Maintain a running maximum of the distinct prime factors count.
4. Complexity Analysis:
   • Time Complexity: For each number n, the prime factorization takes O(√n) time. In the worst case, processing all numbers from left to right leads to O((right – left + 1) * √right) time complexity.
   • Space Complexity: O(1) additional space is used, as we only store counts and temporary variables.

Solution Code

class Solution:
    def maximumPrimeFactors(self, left: int, right: int) -> int:
        max_count = 0
        for num in range(left, right + 1):
            count = 0
            temp = num
            factor = 2
            while factor * factor <= temp:
                if temp % factor == 0:
                    count += 1
                    while temp % factor == 0:
                        temp //= factor
                factor += 1
            if temp > 1:
                count += 1
            if count > max_count:
                max_count = count
        return max_count

Explanation

1. Initialization: Start with max_count set to 0 to keep track of the highest number of distinct prime factors found.
2. Iterate Through Range: For each number num in the range [left, right]:
   • Initialize count to 0 for the current number.
   • Use a temporary variable temp set to num to perform factorization.
   • Check divisibility starting from factor = 2 up to √temp.
3. Prime Factorization:
   • If factor divides temp, increment count (indicating a distinct prime factor) and divide temp by factor until it is no longer divisible.
   • Move to the next factor.
4. Remaining Prime Factor: If after processing all factors up to √temp, temp is greater than 1, it is also a prime factor, so increment count.
5. Update Maximum: Compare count with max_count and update max_count if count is greater.
6. Return Result: After processing all numbers, return max_count, which holds the maximum distinct prime factors for any number in the range.

This approach efficiently checks each number in the range, computes its distinct prime factors, and tracks the maximum count encountered, providing the solution to the problem.