GCD and LCM Euclidean Algorithm in DSA C# - 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 write a simple C# program that calculates the GCD and LCM of two given numbers using the Euclidean algorithm.

📋 What You'll Learn

Create two integer variables named number1 and number2 with exact values 48 and 180.

Create an integer variable named temp to help with calculations.

Write a while loop using the Euclidean algorithm to find the GCD of number1 and number2.

Calculate the LCM using the formula: (number1 * number2) / GCD.

Print the GCD and LCM values exactly as shown.

💡 Why This Matters

🌍 Real World

Finding GCD and LCM helps in tasks like dividing resources evenly or scheduling repeating events.

💼 Career

Understanding Euclidean algorithm and basic loops is essential for programming jobs involving algorithms and problem solving.

1

Create the initial numbers

Create two integer variables called number1 and number2 with values 48 and 180 respectively.

DSA C#

# Create two integers number1 and number2 with values 48 and 180
// Your code here

Hint

Use int to declare variables and assign the exact values.

2

Add a helper variable for calculation

Add an integer variable called temp to help with the Euclidean algorithm calculations.

DSA C#

int number1 = 48;
int number2 = 180;
// Declare an integer variable temp
// Your code here

Hint

Just declare temp as an integer without assigning a value yet.

3

Calculate the GCD using Euclidean algorithm

Write a while loop that runs while number2 != 0. Inside the loop, assign temp = number2, then update number2 = number1 % number2, and finally update number1 = temp. This will calculate the GCD in number1.

DSA C#

int number1 = 48;
int number2 = 180;
int temp;
// Write a while loop to calculate GCD using Euclidean algorithm
// Your code here

Hint

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

4

Calculate and print the GCD and LCM

Calculate the LCM using int lcm = (48 * 180) / number1;. Then print the GCD with Console.WriteLine($"GCD: {number1}"); and the LCM with Console.WriteLine($"LCM: {lcm}");.

DSA C#

int number1 = 48;
int number2 = 180;
int temp;

while (number2 != 0)
{
    temp = number2;
    number2 = number1 % number2;
    number1 = temp;
}
// Calculate LCM and print GCD and LCM
// Your code here

Hint

Use the formula LCM = (number1_original * number2_original) / GCD. Remember to print exactly as shown.

Practice

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

easy

A. The difference between a and b

B. The Greatest Common Divisor (GCD)

C. The sum of a and b

D. The Least Common Multiple (LCM)

Solution

1. Step 1: Understand the Euclidean algorithm purpose

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

2. Step 2: Differentiate GCD from LCM

   While GCD finds the largest number dividing both, LCM finds the smallest number divisible by both. The Euclidean algorithm directly calculates GCD.

3. Final Answer:

   The Greatest Common Divisor (GCD) -> Option B

4. Quick Check:

   Euclidean algorithm = GCD [OK]

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

Common Mistakes:

• Confusing GCD with LCM
• Thinking it calculates sum or difference
• Assuming it finds prime factors directly

2. Which of the following is the correct C# syntax to calculate the GCD of two integers a and b using the built-in method?

easy

A. int gcd = Math.GreatestCommonDivisor(a, b);

B. int gcd = Math.GCD(a, b);

C. int gcd = Math.gcd(a, b);

D. int gcd = Math.Gcd(a, b);

Solution

1. Step 1: Identify the correct method name in C#

   The .NET 6+ Math class provides the method Math.Gcd(int, int) to calculate the greatest common divisor.

2. Step 2: Check method name casing and spelling

   Method names in C# are PascalCase, so Math.Gcd is correct, while others are incorrect or do not exist.

3. Final Answer:

   int gcd = Math.Gcd(a, b); -> Option D

4. Quick Check:

   Correct method is Math.Gcd [OK]

Hint: Use Math.Gcd with capital G and C# PascalCase [OK]

Common Mistakes:

• Using incorrect method names like Math.gcd or Math.GCD
• Assuming method is GreatestCommonDivisor
• Wrong casing causing compile errors

3. What is the output of the following C# code?

int a = 48;
int b = 18;
int gcd = Math.Gcd(a, b);
int lcm = (a * b) / gcd;
Console.WriteLine($"GCD: {gcd}, LCM: {lcm}");

medium

A. GCD: 18, LCM: 48

B. GCD: 12, LCM: 72

C. GCD: 6, LCM: 144

D. GCD: 3, LCM: 288

Solution

1. Step 1: Calculate GCD of 48 and 18

   Using Euclidean algorithm: 48 % 18 = 12, 18 % 12 = 6, 12 % 6 = 0, so GCD = 6.

2. Step 2: Calculate LCM using formula

   LCM = (48 * 18) / 6 = 864 / 6 = 144.

3. Final Answer:

   GCD: 6, LCM: 144 -> Option C

4. Quick Check:

   GCD=6, LCM=144 [OK]

Hint: LCM = (a * b) / GCD always [OK]

Common Mistakes:

• Calculating GCD incorrectly
• Swapping GCD and LCM values
• Forgetting to divide product by GCD for LCM

4. Identify the error in this C# code snippet that tries to compute the GCD using Euclidean algorithm:

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

medium

A. Returns b instead of a after loop ends

B. Uses wrong operator for remainder

C. Infinite loop because b never changes

D. Missing base case for zero inputs

Solution

1. Step 1: Trace the Euclidean algorithm loop

   The loop updates a and b until b becomes zero. At loop end, a holds the GCD.

2. Step 2: Check return statement correctness

   The code returns b, which is zero at loop end. It should return a which holds the GCD.

3. Final Answer:

   Returns b instead of a after loop ends -> Option A

4. Quick Check:

   Return a after loop [OK]

Hint: Return 'a' after loop, not 'b' [OK]

Common Mistakes:

• Returning the wrong variable after loop
• Confusing remainder operator with division
• Not updating variables inside loop

5. Given two arrays A = [12, 15, 21] and B = [18, 35, 14], which C# code snippet correctly computes an array of LCMs for each pair (A[i], B[i]) using the Euclidean algorithm?

hard

A. int[] lcms = new int[A.Length];
for (int i = 0; i < A.Length; i++) {
int gcd = Math.Gcd(A[i], B[i]);
lcms[i] = (A[i] * B[i]) / gcd;
}

B. int[] lcms = new int[A.Length];
for (int i = 0; i < A.Length; i++) {
int gcd = Math.Gcd(A[i], B[i]);
lcms[i] = gcd * A[i] * B[i];
}

C. int[] lcms = new int[A.Length];
for (int i = 0; i < A.Length; i++) {
lcms[i] = Math.Gcd(A[i], B[i]);
}

D. int[] lcms = new int[A.Length];
for (int i = 0; i < A.Length; i++) {
lcms[i] = (A[i] + B[i]) / Math.Gcd(A[i], B[i]);
}

Solution

1. Step 1: Calculate GCD for each pair

   Use Math.Gcd(A[i], B[i]) to find the greatest common divisor for each pair.

2. Step 2: Compute LCM using formula

   LCM = (A[i] * B[i]) / GCD. Store this in the lcms array at index i.

3. Final Answer:

   Option A code snippet correctly computes LCM array -> Option A

4. Quick Check:

   LCM = (a*b)/GCD for each pair [OK]

Hint: LCM = (a*b)/GCD for each pair in loop [OK]

Common Mistakes:

• Multiplying GCD with product instead of dividing
• Assigning GCD instead of LCM to result array
• Using sum instead of product in LCM formula