Java Coding Test Course, Finding Least Common Multiple

In this article, we will explore in depth the method to calculate the Least Common Multiple (LCM) using Java, a problem that frequently appears in coding tests. The Least Common Multiple refers to the smallest multiple of two or more numbers and plays a very important role in various algorithmic problems.

1. Problem Definition

Here is a simple definition of the problem of finding the Least Common Multiple.

        Problem: Given two integers a and b, find the least common multiple of a and b.
        Example:
        Input: a = 4, b = 6
        Output: 12

2. What is Least Common Multiple (L.C.M)?

The Least Common Multiple refers to the smallest multiple among two numbers. For example, the multiples of 4 and 6 are as follows:

  • Multiples of 4: 4, 8, 12, 16, 20, …
  • Multiples of 6: 6, 12, 18, 24, 30, …

In the above example, the smallest number among the multiples of 4 and 6 is 12. Therefore, 12 is the Least Common Multiple of 4 and 6.

3. Common Divisors and Multiples

To understand the Least Common Multiple, it is essential to first understand the Greatest Common Divisor (GCD). The Greatest Common Divisor refers to the largest common divisor of two numbers. The Least Common Multiple can be calculated using the Greatest Common Divisor as follows:

        LCM(a, b) = (a * b) / GCD(a, b)

Using the above formula, we can quickly and efficiently find the Least Common Multiple even for large numbers.

4. Algorithm Design

The algorithm to find the Least Common Multiple can be designed as follows:

  1. Input the two integers a and b.
  2. Calculate GCD(a, b).
  3. Calculate LCM(a, b).
  4. Output the result.

5. Java Code Implementation

Now, let’s implement the above algorithm using Java code.

        public class LCMCalculator {
            // Method to calculate Greatest Common Divisor (GCD)
            public static int gcd(int a, int b) {
                while (b != 0) {
                    int temp = b;
                    b = a % b;
                    a = temp;
                }
                return a;
            }

            // Method to calculate Least Common Multiple (LCM)
            public static int lcm(int a, int b) {
                return (a * b) / gcd(a, b);
            }

            // Main method
            public static void main(String[] args) {
                int a = 4;
                int b = 6;
                
                int result = lcm(a, b);
                System.out.println("Least Common Multiple: " + result);
            }
        }

5.1 Code Explanation

Let’s explain how the above code works.

  1. gcd method: Accepts two integers a and b and calculates the Greatest Common Divisor. It efficiently computes the GCD using the Euclidean algorithm.
  2. lcm method: A method to find the Least Common Multiple of two numbers, applying the previously described GCD formula.
  3. main method: Calculates and outputs the Least Common Multiple for two numbers input by the user.

6. Additional Examples and Tests

Now, let’s test some other inputs. We can enhance the reliability of the code with various test cases.

Example 1

        Input: a = 15, b = 20
        Output: 60

Example 2

        Input: a = 9, b = 12
        Output: 36

Example 3

        Input: a = 7, b = 5
        Output: 35

7. Time Complexity Analysis

The time complexity of the above algorithm can be analyzed as follows:

  • The gcd method has a time complexity of O(log(min(a, b))).
  • Thus, the overall time complexity is O(log(min(a, b))), making it very efficient.

8. Conclusion

In this lecture, we introduced the algorithm for finding the Least Common Multiple using Java. We covered not only the design of the algorithm but also the implementation process and time complexity analysis. Since this problem frequently appears in coding tests, we encourage you to build your skills through sufficient practice. We plan to cover useful algorithms in the next lecture, so stay tuned!