Exercise 2.3.30 (Find all $n$ such that $\phi(2n)>\phi(n)$)

Proof. If $n$ is odd then $\varphi (2n)=\varphi (n)$ by Problem 29. So, if $\varphi (2n)>\varphi (n)$ , then $n$ is even.

Conversely, if $n$ is even, then $n\wedge 2>1$. By Problem 27, $\varphi (2n)>\varphi (2)\varphi (n)=\varphi (n)$.

The set of positive integers satisfying $\varphi (2n)>\varphi (n)$ is the set of even numbers. □