Simplified Fractions - Practice Coding Problems

Simplified Fractions - Problem

Given an integer n, return a list of all simplified fractions between 0 and 1 (exclusive) such that the denominator is less-than-or-equal-to n.

A fraction is simplified if the greatest common divisor (GCD) of the numerator and denominator is 1.

You can return the answer in any order.

Input & Output

Example 1 — Basic Case

$ Input: n = 2

› Output: ["1/2"]

💡 Note: Only one fraction with denominator ≤ 2: 1/2 is already simplified since gcd(1,2) = 1

Example 2 — Multiple Denominators

$ Input: n = 3

› Output: ["1/2", "1/3", "2/3"]

💡 Note: Denominators 2 and 3: 1/2 (gcd=1), 1/3 (gcd=1), 2/3 (gcd=1). All are simplified.

Example 3 — With Non-Simplified

$ Input: n = 4

› Output: ["1/2", "1/3", "1/4", "2/3", "3/4"]

💡 Note: Skip 2/4 because gcd(2,4) = 2, so it simplifies to 1/2 which we already have.

Constraints

1 ≤ n ≤ 100

Visualization

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Understanding the Visualization

Input

Integer n representing maximum denominator

Process

Generate fractions and check if simplified (GCD = 1)

Output

List of simplified fraction strings

Key Takeaway

🎯 Key Insight: A fraction is simplified when its numerator and denominator share no common factors (GCD = 1)

Asked in

G Google 15 M Microsoft 12

The key insight is that a fraction is simplified when gcd(numerator, denominator) = 1. We generate all possible fractions and filter using GCD calculation. Best approach uses efficient Euclidean algorithm. Time: O(n² log n), Space: O(1)

Common Approaches

Approach	Time	Space	Notes
✓ Euler's Totient Function	O(n² log n)	O(1)	Use number theory to generate simplified fractions more efficiently
Brute Force with GCD Check	O(n² log n)	O(1)	Check all possible fractions and keep only simplified ones

Euler's Totient Function — Algorithm Steps

Iterate through denominators from 2 to n
For each denominator, iterate numerators from 1 to denominator-1
Use optimized GCD calculation
Add fraction to result if GCD is 1

Visualization

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Step-by-Step Walkthrough

Generate Fractions

Create fractions systematically

Fast GCD

Use Euclidean algorithm for GCD

Filter Results

Keep only simplified fractions

Code -

solution.c — C

#include <stdio.h>
#include <stdlib.h>
#include <string.h>

void solution(int n) {
    printf("[");
    int first = 1;
    for (int denominator = 2; denominator <= n; denominator++) {
        for (int numerator = 1; numerator < denominator; numerator++) {
            // Optimized GCD calculation
            int a = numerator, b = denominator;
            while (b != 0) {
                int temp = a % b;
                a = b;
                b = temp;
            }
            if (a == 1) {  // GCD is 1
                if (!first) printf(",");
                printf("\"%d/%d\"", numerator, denominator);
                first = 0;
            }
        }
    }
    printf("]\n");
}

int main() {
    int n;
    scanf("%d", &n);
    solution(n);
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(n² log n)

Same asymptotic complexity but with optimized GCD implementation

⚠ Quadratic Growth

Space Complexity

O(1)

Only using variables for computation, result space not counted

✓ Linear Space

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Medium Frequency

~15 min Avg. Time

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Smart Actions

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Input & Output

Constraints

Visualization

Related Problems

Common Approaches

Euler's Totient Function — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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