Count Beautiful Numbers

Problem Description

You are given two positive integers, l and r. A positive integer is called beautiful if the product of its digits is divisible by the sum of its digits. Return the count of beautiful numbers between l and r, inclusive.

Key Insights

• A number is beautiful if the product of its digits divided by the sum of its digits is an integer.
• The problem requires iterating through a potentially large range of numbers (up to 10^9), making a direct approach infeasible.
• Efficiently calculating the digit product and sum is crucial to solving this problem within constraints.

Space and Time Complexity

Time Complexity: O(n * d), where n is the number of integers between l and r, and d is the number of digits in each integer (up to 10). Space Complexity: O(1), as we are using a constant amount of space for calculations.

Solution

To solve the problem, we will iterate through each integer from l to r and calculate the sum and product of its digits. We will check if the product is divisible by the sum. If it is, we increment a counter.

The algorithm involves:

1. Looping through each integer from l to r.
2. For each integer, extract its digits.
3. Calculate the sum and product of these digits.
4. Check if the product is divisible by the sum.
5. Count how many integers satisfy this condition.

This approach utilizes basic arithmetic and digit manipulation to solve the problem effectively.

Code Solutions