GCD and LCM Euclidean Algorithm in DSA Rust - Build from Scratch

GCD and LCM Euclidean Algorithm

📖 Scenario: Imagine you are helping a friend who loves baking. They want to find the greatest common divisor (GCD) and least common multiple (LCM) of two numbers to decide how to evenly divide ingredients or plan baking batches.

🎯 Goal: You will build a small Rust program that calculates the GCD and LCM of two given numbers using the Euclidean algorithm.

📋 What You'll Learn

Create two variables a and b with exact integer values

Create a variable original_a to store the initial value of a

Implement the Euclidean algorithm using a while loop to find the GCD

Calculate the LCM using the formula (original_a * b) / a

Print the GCD and LCM values exactly as specified

💡 Why This Matters

🌍 Real World

Finding GCD and LCM helps in tasks like dividing resources evenly or scheduling repeated events without overlap.

💼 Career

Understanding Euclidean algorithm and basic Rust programming is useful for software developers working on algorithms, embedded systems, or performance-critical applications.

1

Create the initial numbers

Create two variables called a and b with the exact values 48 and 18 respectively.

DSA Rust

# Your code here

Hint

Use let mut to create variables a and b with values 48 and 18.

2

Store the original value of a

Add a variable called original_a that stores the initial value of a.

DSA Rust

fn main() {
    let mut a = 48;
    let mut b = 18;
    // Store original value of a
    // Your code here
}

Hint

Use let original_a = a; to save the starting value of a.

3

Calculate the GCD using Euclidean algorithm

Use a while loop with the condition b != 0 to repeatedly update a and b using the Euclidean algorithm. Inside the loop, create a temporary variable temp to store b, then set b to a % b, and finally set a to temp.

DSA Rust

fn main() {
    let mut a = 48;
    let mut b = 18;
    let original_a = a;
    // Calculate GCD using Euclidean algorithm
    // Your code here
}

Hint

Use the Euclidean algorithm steps inside a while loop to find the GCD.

4

Calculate and print the GCD and LCM

Calculate the LCM using the formula (original_a * b) / a and store it in a variable called lcm. Then print the GCD and LCM using println! exactly as: "GCD: {} and LCM: {} with the values of a and lcm.

DSA Rust

fn main() {
    let mut a = 48;
    let mut b = 18;
    let original_a = a;
    while b != 0 {
        let temp = b;
        b = a % b;
        a = temp;
    }
    // Calculate LCM and print GCD and LCM
    // Your code here
}

Hint

Calculate LCM with (original_a * 18) / a and print both values with println!.

Practice

1. What does the Euclidean algorithm primarily calculate when given two positive integers a and b?

easy

A. The greatest common divisor (GCD) of a and b

B. The least common multiple (LCM) of a and b

C. The sum of a and b

D. The difference between a and b

Solution

1. Step 1: Understand Euclidean algorithm purpose

   The Euclidean algorithm is designed to find the greatest common divisor (GCD) of two integers by repeatedly applying the remainder operation.

2. Step 2: Differentiate GCD from LCM

   While GCD is the largest number dividing both without remainder, LCM is the smallest number divisible by both. Euclidean algorithm directly finds GCD.

3. Final Answer:

   The greatest common divisor (GCD) of a and b -> Option A

4. Quick Check:

   Euclidean algorithm = GCD [OK]

Hint: Euclidean algorithm finds GCD, not LCM or sums [OK]

Common Mistakes:

• Confusing GCD with LCM
• Thinking it calculates sum or difference
• Assuming it returns both GCD and LCM

2. Which of the following Rust function signatures correctly represents a function to compute the GCD of two unsigned integers using the Euclidean algorithm?

easy

A. fn gcd(a: u32, b: u32) -> u32

B. fn gcd(a: i32, b: i32) -> i32

C. fn gcd(a: u32, b: u32) -> bool

D. fn gcd(a: u32, b: u32)

Solution

1. Step 1: Identify correct parameter and return types

   GCD function takes two unsigned integers (u32) and returns an unsigned integer (u32) representing the GCD.

2. Step 2: Check function signature correctness

   fn gcd(a: u32, b: u32) -> u32 matches this: inputs are u32, output is u32. fn gcd(a: i32, b: i32) -> i32 uses signed integers, which is less common for GCD. fn gcd(a: u32, b: u32) -> bool returns bool, incorrect. fn gcd(a: u32, b: u32) lacks return type.

3. Final Answer:

   fn gcd(a: u32, b: u32) -> u32 -> Option A

4. Quick Check:

   Correct signature returns u32 [OK]

Hint: GCD inputs and output are unsigned integers [OK]

Common Mistakes:

• Using signed integers instead of unsigned
• Missing return type in function signature
• Returning wrong type like bool

3. What is the output of the following Rust code snippet?

fn gcd(mut a: u32, mut b: u32) -> u32 {
    while b != 0 {
        let temp = b;
        b = a % b;
        a = temp;
    }
    a
}

fn main() {
    let result = gcd(48, 18);
    println!("{}", result);
}

medium

A. 12

B. 6

C. 18

D. 3

Solution

1. Step 1: Trace Euclidean algorithm steps for gcd(48, 18)

   Start with a=48, b=18. Compute remainder: 48 % 18 = 12. Update a=18, b=12. Next, 18 % 12 = 6. Update a=12, b=6. Then, 12 % 6 = 0. Update a=6, b=0.

2. Step 2: Identify when loop ends and return value

   Loop ends when b=0. Return a=6 as GCD.

3. Final Answer:

   6 -> Option B

4. Quick Check:

   GCD(48,18) = 6 [OK]

Hint: Use remainder steps until second number is zero [OK]

Common Mistakes:

• Returning the last remainder instead of last non-zero divisor
• Mixing up a and b updates
• Stopping loop too early

4. The following Rust function is intended to compute the LCM of two numbers using their GCD. Identify the error.

fn gcd(mut a: u32, mut b: u32) -> u32 {
    while b != 0 {
        let temp = b;
        b = a % b;
        a = temp;
    }
    a
}

fn lcm(a: u32, b: u32) -> u32 {
    a * b / gcd(a, b)
}

fn main() {
    println!("{}", lcm(0, 0));
}

medium

A. Wrong function signature for LCM

B. Incorrect GCD calculation logic

C. Missing return statement in LCM function

D. Division by zero error when both inputs are zero

Solution

1. Step 1: Analyze the LCM call with zero inputs

   The LCM function computes (a * b) / gcd(a, b). The main calls lcm(0, 0), where gcd(0, 0) returns 0, leading to division by zero.

2. Step 2: Verify gcd(0, 0) behavior

   In gcd, when a=0, b=0, the while loop doesn't execute (b == 0), so it returns a = 0.

3. Final Answer:

   Division by zero error when both inputs are zero -> Option D

4. Quick Check:

   LCM divides by gcd; gcd(0,0)=0 causes error [OK]

Hint: Check zero inputs cause gcd=0 leading to division error [OK]

Common Mistakes:

• Ignoring zero inputs causing division by zero
• Assuming gcd never returns zero
• Not validating inputs before LCM calculation

5. Given a list of positive integers [12, 15, 20], which Rust code snippet correctly computes their LCM using the Euclidean algorithm for GCD?

hard

A. let lcm = nums.iter().fold(1, |acc, &x| gcd(acc, x));

B. let lcm = nums.iter().fold(0, |acc, &x| acc + x / gcd(acc, x));

C. let lcm = nums.iter().fold(1u32, |acc, &x| acc * x / gcd(acc, x));

D. let lcm = nums.iter().fold(1, |acc, &x| acc * x * gcd(acc, x));

Solution

1. Step 1: Understand LCM folding logic

   LCM of multiple numbers can be found by folding with formula: lcm(a,b) = a * b / gcd(a,b). Starting with 1u32, fold applies this stepwise.

2. Step 2: Analyze each option

   let lcm = nums.iter().fold(1u32, |acc, &x| acc * x / gcd(acc, x)); correctly folds with acc * x / gcd(acc, x). let lcm = nums.iter().fold(0, |acc, &x| acc + x / gcd(acc, x)); uses addition, incorrect. let lcm = nums.iter().fold(1, |acc, &x| gcd(acc, x)); returns gcd only, not LCM. let lcm = nums.iter().fold(1, |acc, &x| acc * x * gcd(acc, x)); multiplies by gcd, incorrect.

3. Final Answer:

   let lcm = nums.iter().fold(1u32, |acc, &x| acc * x / gcd(acc, x)); -> Option C

4. Quick Check:

   Fold with acc * x / gcd(acc, x) computes LCM [OK]

Hint: Fold with acc * x / gcd(acc, x) to get LCM of list [OK]

Common Mistakes:

• Using addition instead of multiplication
• Returning gcd instead of LCM
• Multiplying by gcd instead of dividing