Exercise 2.3.29 (Find all $n$ such that $\phi(2n) = \phi(n)$.)

Proof. Problem 28 shows that $\varphi (2n)=\varphi (n)$ implies $n$ is odd.

Conversely, if $n$ is odd, then $n\wedge 2=1$, so

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