GCD and LCM Euclidean Algorithm in DSA Rust - Practice Problems & Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

What is the output of this Rust code for GCD calculation?

Consider the following Rust function that calculates the Greatest Common Divisor (GCD) using the Euclidean algorithm. What will be printed when calling gcd(48, 18)?

DSA Rust

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

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

A 18

B 12

C 6

D 3

💡 Hint

Remember the Euclidean algorithm repeatedly replaces the larger number by the remainder until the remainder is zero.

The Euclidean algorithm finds the GCD by repeatedly taking remainders. gcd(48,18) reduces as 48%18=12, then gcd(18,12), 18%12=6, then gcd(12,6), 12%6=0, so GCD is 6.

❓ Predict Output

intermediate

What is the output of this Rust code for LCM calculation?

Given the following Rust code that calculates the Least Common Multiple (LCM) using the GCD function, what will be printed when calling lcm(21, 6)?

DSA Rust

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(21, 6));
}

A 126

B 42

C 18

D 12

💡 Hint

LCM can be found by multiplying the numbers and dividing by their GCD.

GCD of 21 and 6 is 3. LCM = (21*6)/3 = 126/3 = 42.

🧠 Conceptual

advanced

Why does the Euclidean algorithm for GCD always terminate?

The Euclidean algorithm repeatedly replaces the pair (a, b) with (b, a % b) until b becomes zero. Why does this process always end?

A Because the remainder a % b is always smaller than b, so the numbers decrease until zero.

B Because the algorithm uses recursion which has a fixed depth limit.

C Because the numbers are converted to floating point which eventually rounds to zero.

D Because the algorithm adds numbers until they reach a maximum value.

💡 Hint

Think about how the remainder changes each step.

At each step, the remainder is strictly less than the previous divisor, so the numbers decrease and eventually reach zero, ending the loop.

🔧 Debug

advanced

What error does this Rust code produce when calculating GCD?

Consider this Rust code snippet intended to compute the GCD. What error will it produce when compiled?

DSA Rust

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

fn main() {
    println!("{}", gcd(10, 5));
}

A No error, outputs 5

B Runtime panic due to division by zero

C Logical error producing wrong GCD

D SyntaxError due to missing semicolon

💡 Hint

Check punctuation carefully in Rust code.

The line 'let temp = a % b' is missing a semicolon, causing a syntax error.

🚀 Application

expert

What is the value of variable 'result' after this Rust code runs?

This Rust code calculates the LCM of three numbers using the GCD function. What is the final value of 'result'?

DSA Rust

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() {
    let a = 4;
    let b = 6;
    let c = 8;
    let result = lcm(lcm(a, b), c);
    println!("{}", result);
}

A 24

B 96

C 48

D 12

💡 Hint

Calculate LCM step by step: first lcm(a,b), then lcm of that with c.

LCM(4,6) = 12. Then LCM(12,8) = (12*8)/GCD(12,8) = 96/4 = 24. So final result is 24.

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