Euclidean Algorithm in Java

The Euclidean algorithm or the algorithm of descent is a well-established method of mathematics that is applied to find the GCD. The GCD stands for the largest equal divisor and is a positive integer. It divides both numbers without a remainder. Its use is essential in number theory, cryptography, computing, and a lot more.

The algorithm is called Euclidean distance because it was introduced by Greek mathematician Euclid in Elements about 300 BC. The principle that the algorithm of Euclid uses is that the GCD of two numbers also divides the difference between these two numbers.

In the algorithm, the dietic process consists of substituting the most significant number by the quotient that results when the greater number is divided by the smaller number. This process proceeds until one of the numbers becomes zero; the other number then is the GCD of this pair of numbers.

It reflects how one of the algorithm's main advantages is that it is easy and fast to implement. Due to its ability, it can easily find out the great GCD precisely at a faster rate, even for a huge number, making it essential in any computing application.

It is suitable for the iterative or recursive implementation of the algorithm that essentially involves simple mathematical computations. Hence, the practicality and the historical aspect of it make the Euclidean algorithm an essential component of learning mathematics and computer science.

Euclidean Algorithm

Initial Setup: If the numbers a and b are given and a is larger than the number b.

Check for Base Case: If b == 0, that means that we have found the GCD and the value of the GCD is a. The algorithm terminates.

Recursive Step: If b!= 0, replace a with b and b with a % b (the remainder of the division of a by b).

Repeat: Goto Step 2 and step 3 until b becomes 0.

1. Iterative Approach

The iteration means the usage of loops that enforce the repetition of a perticular set of codes until a condition is met. It is especially beneficial when a problem has been reduced to a sequence of steps to its solution.

File Name: EuclideanAlgorithm.java

 
public class EuclideanAlgorithm {
    public static int gcdIterative(int a, int b) {
        // Continue looping until the second number (b) becomes zero
        while (b != 0) {
            int temp = b; // Store the current value of b in a temporary variable
            b = a % b;    // Update b to the remainder of a divided by b
            a = temp;     // Update a to the previous value of b
        }
        // When b becomes zero, a contains the GCD
        return a;
    }
    public static void main(String[] args) {
        int a = 48; // First integer for GCD calculation
        int b = 18; // Second integer for GCD calculation
        // Call the gcdIterative method to compute the GCD
        int gcd = gcdIterative(a, b);
        System.out.println("GCDs of " + a + " and " + b + " is: " + gcd);
    }
}   

Output:

 
GCDs of 48 and 18 is: 6

2. Recursive Approach

A recursive technique uses a method to call another method of the same type but to solve a simpler version of the problem. The approach is elegant for problems that can be divided into subproblems, and all these subproblems are of the same type.

File Name: EuclideanAlgorithm.java

 
public class EuclideanAlgorithm {
    public static int gcdRecursive(int a, int b) {
        // Base case: if b is 0, return a as the GCD
        if (b == 0) {
            return a;
        }
        // Recursive call: calculate GCD of b and the remainder of a divided by b
        return gcdRecursive(b, a % b);
    }
    public static void main(String[] args) {
        int a = 48; // First integer for GCD calculation
        int b = 18; // Second integer for GCD calculation
        // Call the gcdRecursive method to compute the GCD
        int gcd = gcdRecursive(a, b);
        System.out.println("GCDs of " + a + " and " + b + " is: " + gcd);
    }
}   

Output:

 
GCDs of 48 and 18 is: 6

Conclusion

The Euclidean algorithm studied here iteratively or recursively is a basic and efficient procedure to find out the GCD of two integers. Because of its simplicity, elegance and computational efficiency, it is considered a key instrument in both mathematics and computer science.

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