C# Coding Test Course, Euclidean Algorithm

Introduction

One of the algorithms that frequently appears in coding tests, the Euclidean Algorithm, is an efficient method for finding the greatest common divisor (GCD) of two numbers.
This method was introduced by the ancient Greek mathematician Euclid and is a fundamental concept that must be understood regardless of the programming language.
In this course, we will learn how to implement the Euclidean Algorithm using C#.

Problem Description

The problem is to find the greatest common divisor (GCD) of two given integers a and b.
The GCD is the largest positive divisor that both integers share.
For example, when a = 48 and b = 18, GCD(48, 18) = 6.

Input Format

The first line contains two integers a and b separated by a space (1 ≤ a, b ≤ 10^9).

Output Format

The output is the greatest common divisor (GCD) of the two numbers a and b.

Understanding the Euclidean Algorithm

The Euclidean Algorithm is based on the following principle:

1. Given two numbers a and b, let r be the remainder of a divided by b, then GCD(a, b) = GCD(b, r).
2. By repeating this process until b becomes 0, a will become the GCD.

This method is very efficient and can be computed quickly even when a and b are large.
Due to this property, the Euclidean Algorithm is useful in many algorithmic problems.

Implementing the Euclidean Algorithm in C#

Now, let’s implement the Euclidean Algorithm using C#. Below is the code that implements this algorithm.

    using System;

    class Program
    {
        static void Main(string[] args)
        {
            // Get input values
            string[] inputs = Console.ReadLine().Split(' ');
            int a = int.Parse(inputs[0]);
            int b = int.Parse(inputs[1]);

            // Calculate the greatest common divisor
            int gcd = EuclideanGCD(a, b);
            Console.WriteLine(gcd);
        }

        static int EuclideanGCD(int a, int b)
        {
            while (b != 0)
            {
                int temp = b;
                b = a % b;  // Find the remainder
                a = temp;   // Assign the value of b to a
            }
            return a;  // Return the GCD
        }
    }
    

Code Explanation

The code above takes two integers as input and uses the Euclidean Algorithm to calculate the GCD.
In the Main method, it receives two numbers from the user and calls the EuclideanGCD method to compute the GCD.
The EuclideanGCD method uses a while loop to repeat the process until b becomes 0, while calculating the GCD.

Time Complexity and Space Complexity

The time complexity of the Euclidean Algorithm is O(log(min(a, b))). This is because the size of the two numbers decreases exponentially.
The space complexity is O(1), as it does not use any additional data structures, making it very efficient.

Test Cases

You can use the following test cases to verify the accuracy of the algorithm.

    // Test cases
    48 18  // Expected output: 6
    56 98  // Expected output: 14
    101 103 // Expected output: 1
    1000000000 500000000 // Expected output: 500000000
    

Conclusion

The Euclidean Algorithm is a very efficient method for finding the greatest common divisor.
Through this article, we explored how to implement the Euclidean Algorithm using C#.
Since you will often encounter such algorithm problems in coding tests, it is important to understand and practice them.

We plan to conduct in-depth courses on various algorithms in the future, so please stay tuned!