Java Coding Test Course, Implementing the Euler's Totient Function

In modern society where online coding tests are increasingly required, the ability to effectively solve algorithm problems is an essential skill for developers. Today, we will deal with an algorithm problem that utilizes mathematical concepts. In this course, we will implement the Euler’s Totient Function and explain the process of solving it using the Java programming language.

What is Euler’s Totient Function?

The Euler’s Totient Function for a given positive integer n represents the number of integers between 1 and n that are coprime with n. In other words, it counts the number of integers whose greatest common divisor with n is 1. For example:

φ(1) = 1 (The only number coprime with 1 is 1 itself)
φ(2) = 1 (The only number coprime with 2 among 1 and 2 is 1)
φ(3) = 2 (The numbers coprime with 3 among 1, 2, and 3 are 1 and 2)
φ(4) = 2 (The numbers coprime with 4 among 1, 2, 3, and 4 are 1 and 3)

This function is also important in various fields of number theory and cryptography. Now, let’s solve the problem of implementing this function.

Problem Description

Implement a function that calculates Euler’s Totient Function for a given integer n. The input to the function is an integer n (1 ≤ n ≤ 10^6), and the output is the value of φ(n).

Problem Solving Strategy

There are various methods to find Euler’s Totient Function, but we can implement it efficiently using the following algorithm:

Prime Factorization: First, find the prime factors of n. Prime factors are the smallest primes that can express an integer as a product.
Calculating the Euler’s Totient Function: It can be calculated using the property of π(n). This can be expressed in the following formula:

φ(n) = n * (1 - 1/p1) * (1 - 1/p2) * ... * (1 - 1/pk)

In this formula, p1, p2, ..., pk are the prime factors of n. This formula allows us to compute Euler’s Totient Function quickly.

Java Code Implementation

Now let’s write a Java code to implement the Euler’s Totient Function. The following code snippet shows how to solve this problem:

public class EulerTotient {

    public static int eulerTotient(int n) {
        int result = n; // Initialize the result to n
        for (int p = 2; p * p <= n; p++) {
            // Check if p is a prime factor of n
            if (n % p == 0) {
                // If a prime factor, calculate the result
                while (n % p == 0) {
                    n /= p;
                }
                result -= result / p; // Update the result
            }
        }
        // Handle the last prime factor
        if (n > 1) {
            result -= result / n;
        }
        return result;
    }

    public static void main(String[] args) {
        int n = 10; // Define the input for testing
        System.out.println("φ(" + n + ") = " + eulerTotient(n)); // Output the result
    }
}

The above code performs operations using prime factorization to efficiently calculate the Euler’s Totient Function. For the input value n, π(n) is calculated and the result is printed.

Code Explanation

1. Function Definition

public static int eulerTotient(int n) method calculates and returns the Euler’s Totient Function for the input integer n. The final result is stored in a variable called result, initialized to n.

2. Prime Factor Check

A for loop is used where p ranges from 2 to √n, checking whether p is a prime factor of n. If it is a prime factor, it performs operations on all powers of p dividing n.

3. Result Calculation

When a prime factor p is found, the result is updated considering the relation with n to ultimately calculate the value of φ(n). Lastly, if n is greater than 1, we must also handle the last prime factor.

4. Main Method

The main method defines the test value for n and is structured to output the result.

Result Check

When the above code is executed, you can verify the value of the Euler’s Totient Function for the input value 10. The result is as follows:

φ(10) = 4

There are 4 numbers less than 10 that are coprime with 10, namely 1, 3, 7, and 9. This confirms that the implemented Euler’s Totient Function works correctly.

Review and Conclusion

In this course, we have implemented the Euler’s Totient Function and explored the process of solving algorithm problems. The Euler’s Totient Function is an algorithm based on mathematical reasoning, and it’s beneficial to know it due to its large applicability in solving various problems. The code implemented in Java can efficiently calculate Euler’s Totient values for numbers within a specific range and can be easily applied in various situations.

Continuing to tackle algorithm problems and practicing repeatedly will be important moving forward. This will greatly help in achieving results in coding tests. Thank you for reading!

Java Coding Test Course, Implementing the Euler’s Totient Function