GCD and LCM Euclidean Algorithm in DSA Java - 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 main idea behind the Euclidean Algorithm for finding GCD?

The Euclidean Algorithm finds the GCD by repeatedly replacing the larger number by the remainder when the larger number is divided by the smaller number, until the remainder is zero. The last non-zero remainder is the GCD.

beginner

How is LCM related to GCD for two numbers a and b?

LCM (Least Common Multiple) and GCD are related by the formula: LCM(a, b) = (a × b) / GCD(a, b). This means once you know the GCD, you can find the LCM easily.

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

Why is the Euclidean Algorithm efficient compared to checking all divisors?

The Euclidean Algorithm uses division and remainders to reduce the problem size quickly, avoiding checking all possible divisors. This makes it much faster especially for large numbers.

What is the GCD of 48 and 18 using the Euclidean Algorithm?

A 6

B 12

C 18

D 3

48 % 18 = 12, 18 % 12 = 6, 12 % 6 = 0, so GCD is 6.

Which step ends the Euclidean Algorithm?

A When numbers are equal

B When remainder is zero

C When one number is 1

D When both numbers are prime

The algorithm stops when the remainder becomes zero.

If GCD(15, 25) = 5, what is LCM(15, 25)?

A 125

B 50

C 75

D 30

LCM = (15 × 25) / 5 = 375 / 5 = 75.

What is the first remainder when finding GCD of 56 and 15?

A 7

B 15

C 1

D 11

56 % 15 = 11, so first remainder is 11.

Which of these is NOT true about the Euclidean Algorithm?

A It uses subtraction only

B It uses division and remainder

C It reduces problem size each step

D It finds GCD efficiently

The Euclidean Algorithm uses division and remainder, not just subtraction.

Explain how the Euclidean Algorithm finds the GCD of two numbers step-by-step.

Think about dividing and replacing numbers until remainder is zero.

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

Use multiplication and division with GCD.

Practice

1. What does the Euclidean algorithm primarily calculate between two numbers a and b?

easy

A. The sum of the numbers

B. The Least Common Multiple (LCM)

C. The Greatest Common Divisor (GCD)

D. The difference between the numbers

Solution

1. Step 1: Understand the 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 and others

   While LCM is related to GCD, the Euclidean algorithm directly computes GCD, not LCM or sums/differences.

3. Final Answer:

   The Greatest Common Divisor (GCD) -> Option C

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 finds prime factors directly

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

easy

A. public static int gcd(int a, int b)

B. public int gcd(int a, int b, int c)

C. static void gcd(int a, int b)

D. public static int gcd()

Solution

1. Step 1: Check method parameters and return type

   The method should take two integers as input and return an integer representing the GCD.

2. Step 2: Validate method signature correctness

   public static int gcd(int a, int b) matches this: it is static, returns int, and takes two int parameters. Others have wrong parameters or return type.

3. Final Answer:

   public static int gcd(int a, int b) -> Option A

4. Quick Check:

   Correct signature = public static int gcd(int a, int b) [OK]

Hint: GCD method needs two int inputs and int return [OK]

Common Mistakes:

• Adding extra parameters unnecessarily
• Using void return type instead of int
• Omitting static keyword when needed

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

public class Test {
  public static int gcd(int a, int b) {
    if (b == 0) return a;
    return gcd(b, a % b);
  }
  public static void main(String[] args) {
    System.out.println(gcd(48, 18));
  }
}

medium

A. 6

B. 12

C. 18

D. 3

Solution

1. Step 1: Trace gcd(48, 18) calls

   gcd(48,18) calls gcd(18, 48%18=12), then gcd(18,12) calls gcd(12, 18%12=6), then gcd(12,6) calls gcd(6, 12%6=0), returns 6.

2. Step 2: Final return value

   When second parameter is 0, return first parameter which is 6.

3. Final Answer:

   6 -> Option A

4. Quick Check:

   GCD(48,18) = 6 [OK]

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

Common Mistakes:

• Stopping too early without full recursion
• Mixing up modulo operation
• Returning wrong variable when b=0

4. Identify the error in the following Java method for computing LCM using GCD and fix it:

public static int lcm(int a, int b) {
  return a * b / gcd(a, b);
}

public static int gcd(int a, int b) {
  while (b != 0) {
    int temp = a % b;
    a = b;
    b = temp;
  }
  return a;
}

medium

A. Return b instead of a in gcd method

B. gcd method should be static void

C. No error; code is correct

D. Use Math.abs(a * b) to avoid negative LCM

Solution

1. Step 1: Analyze LCM calculation

   LCM is calculated as (a * b) / GCD(a, b). If a or b is negative, product can be negative, causing wrong LCM.

2. Step 2: Fix negative product issue

   Use Math.abs(a * b) to ensure LCM is always positive.

3. Final Answer:

   Use Math.abs(a * b) to avoid negative LCM -> Option D

4. Quick Check:

   LCM must be positive; abs fixes sign [OK]

Hint: Use absolute value for product in LCM formula [OK]

Common Mistakes:

• Ignoring negative inputs causing wrong LCM
• Changing gcd return type incorrectly
• Returning wrong variable in gcd method

5. Given an array of integers nums = {12, 15, 20}, which Java code snippet correctly computes the LCM of all elements using the Euclidean algorithm?

hard

A. int lcm = 1; for(int i=0; i

B. int lcm = nums[0]; for(int i=1; i

C. int lcm = nums[0]; for(int i=1; i

D. int lcm = 0; for(int i=0; i

Solution

1. Step 1: Understand LCM of multiple numbers

   LCM of multiple numbers can be found by iteratively computing LCM of pairs: lcm(a,b,c) = lcm(lcm(a,b), c).

2. Step 2: Check code correctness

   int lcm = nums[0]; for(int i=1; i

3. Final Answer:

   int lcm = nums[0]; for(int i=1; i -> Option B

4. Quick Check:

   Iterative LCM with gcd formula works [OK]

Hint: Iterate LCM pairwise using lcm = (a*b)/gcd(a,b) [OK]

Common Mistakes:

• Using gcd instead of lcm in loop
• Summing elements instead of computing LCM
• Starting lcm from 0 causing division errors