GCD and LCM Euclidean Algorithm in DSA Rust - Cheat Sheet & Quick Revision

Recall & Review

beginner

What does GCD stand for and what does it represent?

GCD stands for Greatest Common Divisor. It is the largest number that divides two or more numbers without leaving a remainder.

beginner

What is the Euclidean Algorithm used for?

The Euclidean Algorithm is a method to find the GCD of two numbers by repeatedly subtracting the smaller number from the larger one or using the remainder operation until the remainder is zero.

intermediate

How is LCM related to GCD?

LCM (Least Common Multiple) and GCD are related by the formula: LCM(a, b) = (a * b) / GCD(a, b). This means the product of two numbers equals the product of their GCD and LCM.

beginner

What is the base case in the Euclidean Algorithm for GCD?

The base case is when the second number becomes zero. At this point, the first number is the GCD.

intermediate

Show the step-by-step Euclidean Algorithm to find GCD of 48 and 18.

Step 1: 48 % 18 = 12
Step 2: 18 % 12 = 6
Step 3: 12 % 6 = 0
GCD is 6.

What is the GCD of 24 and 36 using the Euclidean Algorithm?

A 6

B 12

C 24

D 18

36 % 24 = 12, 24 % 12 = 0, so GCD is 12.

Which operation is repeatedly used in the Euclidean Algorithm?

A Multiplication

B Subtraction

C Addition

D Modulo (remainder)

The Euclidean Algorithm uses the modulo operation to find the remainder until it reaches zero.

If GCD(8, 12) = 4, what is LCM(8, 12)?

A 24

B 48

C 36

D 16

LCM = (8 * 12) / 4 = 96 / 4 = 24.

What is the first step in the Euclidean Algorithm for GCD(56, 15)?

A 56 % 15 = 11

B 15 % 56 = 15

C 56 - 15 = 41

D 15 - 56 = -41

We calculate 56 % 15 which equals 11 as the first step.

When does the Euclidean Algorithm stop?

A When both numbers are equal

B After a fixed number of steps

C When the remainder becomes zero

D When the smaller number is zero

The algorithm stops when the remainder is zero; the other number at that point is the GCD.

Explain how the Euclidean Algorithm finds the GCD of two numbers.

Think about dividing and taking remainders repeatedly.

Describe the relationship between GCD and LCM and how to calculate LCM using GCD.

Consider how multiplication of numbers relates to their divisors and multiples.

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