Let's tackle the problems in the image one by one.

Problem 2

Given: $p$ is any prime, and $k$ is any positive integer, what is $\varphi(p^k)$?

The Euler's totient function $\varphi(n)$ for $n = p^k$ where $p$ is a prime and $k$ is a positive integer is given by: $\varphi(p^k) = p^k - p^{k-1}$

So the correct answer is: $p^k - p^{k-1}$

Problem 3

Statement: For $n \geq 3$, the value $\varphi(n)$ is even.

Let's consider $n \geq 3$. If $n$ is even, then $\varphi(n)$ must be even since it counts the number of integers up to $n$ that are coprime with $n$. If $n$ is odd, $\varphi(n)$ is also even because the number of integers less than $n$ coprime with $n$ would include all numbers not having the prime factors of $n$. Hence, $\varphi(n)$ is always even.

So the statement is: $\text{True}$

Problem 4

Given:

• $n$ and $b$ are positive integers for which $n$ is odd,
• $a \equiv b \pmod{\varphi(n)}$,

We need to determine whether $2^a \equiv 2^b \pmod{n}$.

From the properties of exponents and modular arithmetic, if $a \equiv b \pmod{\varphi(n)}$, then $2^a \equiv 2^b \pmod{n}$.

So the statement is: $\text{True}$

Do you have any questions or need any further details on these problems?

Here are 5 additional questions you might be interested in:

1. How do you compute the Euler's totient function for any given integer?
2. What are the properties of the Euler's totient function?
3. Can the Euler's totient function be used to solve congruence problems involving large exponents?
4. What are some common applications of the Euler's totient function in number theory?
5. How does the Chinese Remainder Theorem relate to the Euler's totient function?

Tip:

When working with modular arithmetic, always remember the properties of congruences and how they can simplify complex exponentiation problems.