To solve the "Simplified Fractions" problem, identify all fractions between 0 and 1 where the denominator is less than or equal to n. Ensure that each fraction is fully simplified by checking that the greatest common divisor (GCD) of the numerator and denominator is 1. This problem requires a mix of math and string manipulation to generate the results.
Problem Statement
Given an integer n, return all simplified fractions between 0 and 1 (exclusive) where the denominator is less than or equal to n. Each fraction should be in the form of 'numerator/denominator' and you can return the list in any order.
A fraction is considered simplified if the greatest common divisor (GCD) of its numerator and denominator is 1. For example, '2/4' is not simplified because it can be reduced to '1/2'.
Examples
Example 1
Input: n = 2
Output: ["1/2"]
"1/2" is the only unique fraction with a denominator less-than-or-equal-to 2.
Example 2
Input: n = 3
Output: ["1/2","1/3","2/3"]
Example details omitted.
Example 3
Input: n = 4
Output: ["1/2","1/3","1/4","2/3","3/4"]
"2/4" is not a simplified fraction because it can be simplified to "1/2".
Constraints
- 1 <= n <= 100
Solution Approach
Loop through possible fractions
For each denominator from 2 to n, iterate through all possible numerators (from 1 to denominator-1). Check if the GCD of the numerator and denominator is 1 to determine if the fraction is simplified.
Use GCD to simplify fractions
To efficiently check if a fraction is simplified, use the greatest common divisor (GCD) of the numerator and denominator. If the GCD is 1, the fraction is simplified.
Generate result as strings
For each valid simplified fraction, convert the numerator and denominator into a string of the form 'numerator/denominator' and append it to the result list.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
Time complexity depends on the number of fractions generated, which is approximately O(n^2) due to the nested loops. For each fraction, checking the GCD takes constant time, and space complexity is also O(n^2) to store the fractions.
What Interviewers Usually Probe
- Understanding of GCD and its application to fraction simplification
- Ability to work with loops and basic string formatting
- Familiarity with time and space complexity analysis in mathematical problems
Common Pitfalls or Variants
Common pitfalls
- Forgetting to check the GCD of the numerator and denominator to ensure the fraction is simplified
- Generating fractions in the wrong order or including fractions like '2/4' that aren't simplified
- Not handling edge cases, such as when n is 1 (which would not result in any simplified fractions)
Follow-up variants
- Modify the problem to return fractions in ascending order by value
- Return only fractions where the numerator is prime
- Optimize the solution by using a sieve approach to identify valid fractions
How GhostInterview Helps
- GhostInterview provides an efficient way to implement the logic for generating and simplifying fractions with a step-by-step approach.
- It helps quickly identify edge cases and optimize solutions through practical hints.
- With GhostInterview, you can simulate the process and improve your understanding of the mathematical patterns behind fraction simplification.
Topic Pages
Related GhostInterview Pages
- LeetCode Interview Copilot - Use GhostInterview as a live solver when you want direct help with LeetCode-style coding questions.
- Coding Interview Assistant - See how GhostInterview supports array, string, linked list, graph, and tree interview workflows.
- How GhostInterview Works - Review the screenshot, reasoning, and answer flow before using the solver in a live interview.
FAQ
How can I generate all simplified fractions for a given n?
Loop through each denominator from 2 to n and check every possible numerator. Use the GCD to determine if the fraction is simplified.
What is a simplified fraction?
A fraction is simplified if the greatest common divisor (GCD) of the numerator and denominator is 1.
What is the time complexity of solving this problem?
The time complexity is approximately O(n^2) due to the nested loops and GCD checks for each fraction.
What is the primary pattern in this problem?
The primary pattern involves math (GCD and fractions) combined with string manipulation to output the result.
How do I handle fractions like '2/4' that are not simplified?
By checking the GCD of the numerator and denominator, you can avoid adding unsimplified fractions to the result.
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Find the kth factor of n by scanning divisors carefully or using factor pairs to skip unnecessary checks.
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