Help me with this question please it's for java

Question

The totient function \(\phi(n)\) is the count of numbers from \(1\) to \(n\) that are relatively prime with \(n\), that is, have no common divisors with \(n\) except \(1\). For example, 1, 5, 7, and 11 are relatively prime with 12, while 2, 3, 4, 6, 8, 9, and 10 have a common divisor with 12, so \(\phi(12)=4\). This function was introduced by Leonhard Euler (1707-1783), one of the greatest mathematicians of all times. It is used in number theory and in certain algorithms. The totient function is traditionally written using the Greek letter \(\phi\) (phi).

Euler also described a simple way to calculate \(\phi(n)\). As we know, any integer \(n \gt 1\) can be represented as the product of distinct prime factors, raised to the appropriate powers. For example, \(12=2\cdot 2 \cdot 3=2^2 \cdot 3^1\) and \(200 = 2 \cdot 2 \cdot 2 \cdot 5 \cdot 5 = 2^3 \cdot 5^2\). Euler proved that if \(n=p_1^{e_1} \cdot p_2^{e_2} \dots p_k^{e_k}\), then \(\phi(n) = (p_1-1)p_1^{e_1-1} \cdot (p_2-1)p_2^{e_2-1} \dots (p_k-1)p_k^{e_k-1}\). For example, \(\phi(12)=(2-1)\cdot 2^1\cdot(3-1)\cdot 3^0=4\) and \(\phi(200)=(2-1)\cdot 2^2 \cdot (5-1)\cdot5^1 = 80\).

In this question you will write two methods that help calculate \(\phi(n)\) using Euler's formula.

Write a method smallestPrimeFactor(int n) that returns the smallest prime factor of . Note that the smallest prime factor of n is simply the smallest integer greater than or equal to 2 that divides n.

Complete the method smallestPrimeFactor below.

Answer 1

Sure, here's an implementation of the smallestPrimeFactor method in Java:

public static int smallestPrimeFactor(int n) {

   for (int i = 2; i <= n; i++) {

       if (n % i == 0) {

           return i;

       }

   }

   return -1; // indicates n is a prime number

}

This method takes an integer n as input and returns the smallest prime factor of n. It does this by iterating over all integers from 2 to n and checking if they divide n without leaving a remainder. The first integer that satisfies this condition is the smallest prime factor of n.

Note that if n is itself a prime number, the method returns -1 to indicate that there is no smaller prime factor.

I hope this helps! Let me know if you have any further questions.