JavaScript Coding Test Course, Euclidean Algorithm

Hello! Today, we will learn about the algorithm for finding the Greatest Common Divisor (GCD) using the Euclidean algorithm. The Euclidean algorithm is a classical approach to find the GCD of two integers a and b based on the following principles. In this tutorial, we will explore the theory of the Euclidean algorithm and how to implement it in JavaScript step by step.

1. Overview of the Euclidean Algorithm

The Euclidean algorithm is a method proposed by the Greek mathematician Euclid around 300 BC to find the greatest common divisor of two natural numbers a and b. The algorithm works as follows:

  • If b is 0, then GCD(a, b) is a.
  • Otherwise, GCD(a, b) = GCD(b, a mod b).

Here, mod refers to the modulus operation, where a mod b returns the remainder when a is divided by b.

1.1 Example of the Algorithm

For example, if a = 48 and b = 18, we can find the GCD through the following steps:

GCD(48, 18)
= GCD(18, 48 mod 18)
= GCD(18, 12)
= GCD(12, 6)
= GCD(6, 0)
= 6

Therefore, GCD(48, 18) = 6.

2. Implementing the Euclidean Algorithm in JavaScript

Now let’s implement the Euclidean algorithm in JavaScript. Below is the code that implements a function to find the GCD:


function gcd(a, b) {
    if (b === 0) {
        return a;
    }
    return gcd(b, a % b);
}

// Example usage
const a = 48;
const b = 18;
console.log(`GCD of ${a} and ${b} is ${gcd(a, b)}`);

2.1 Explanation of the Code

  • function gcd(a, b): This is a function that takes two arguments a and b and calculates the GCD.
  • if (b === 0): If b is 0, it returns a. This is the basis of the Euclidean algorithm.
  • return gcd(b, a % b): This recursively calls the gcd function, swapping a with b and b with a mod b.

3. Various Uses and Applications

The Euclidean algorithm has various applications in different fields. For example:

  • Solving mathematical problems: It is used not only to find the GCD of two numbers but also to find the GCD of multiple numbers.
  • Computer Science: It is used to compute the GCD required for expressing fractions in their simplest form.
  • Cryptography: The GCD is important in RSA encryption algorithms.

4. Problem Solving: Finding the GCD of Two Numbers

Let’s solve the following problem:


Problem: Write a function that takes two integers and outputs their greatest common divisor.
Input: Two integers a, b (1 <= a, b <= 10000)
Output: The greatest common divisor of a and b

4.1 Problem Solving Process

  1. Input two integers.
  2. Calculate the GCD using the Euclidean algorithm.
  3. Output the calculated GCD.

Now let’s write the complete code as follows:


function gcd(a, b) {
    if (b === 0) {
        return a;
    }
    return gcd(b, a % b);
}

// Getting input from the user
const a = parseInt(prompt("Please enter the first integer: "));
const b = parseInt(prompt("Please enter the second integer: "));

console.log(`GCD of ${a} and ${b} is ${gcd(a, b)}`);

5. Conclusion

In conclusion, we have learned about the JavaScript algorithm using the Euclidean algorithm. I hope this helps you gain a basic understanding of algorithm problems by understanding and implementing the algorithm. If you have any further questions, feel free to leave a comment!

6. Additional Resources

If you want to study the Euclidean algorithm in depth, please refer to the following resources:

Thank you!