Solving $\varphi(x)=x/2$, $\varphi(x)=x/3$ and $\varphi(x)=x/4$

This is Problem 4.141 of N. B. Alfutova and A. V. Ustinov, Algebra and Number Theory (MCCME, 2002), Chapter 4, §4 “Theorems of Fermat and Euler”. Here $\varphi$ is Euler's function. The problem asks to solve, in natural numbers $x$, the equations (a) $\varphi(x)=x/2$; (b) $\varphi(x)=x/3$; (c) $\varphi(x)=x/4$. The book's answers: (a) $x=2^{\alpha}$; (b) $x=2^{\alpha}3^{\beta}$ with $\alpha,\beta\ge1$; (c) no solutions.

Theorem. For natural numbers $x\ge1$:

1. $\varphi(x)=\dfrac{x}{2}$ if and only if $x=2^{\alpha}$ for some integer $\alpha\ge1$;
2. $\varphi(x)=\dfrac{x}{3}$ if and only if $x=2^{\alpha}3^{\beta}$ for some integers $\alpha\ge1$, $\beta\ge1$;

The problem illustrates the product formula $\varphi(x)/x=\prod_{p\mid x}(1-1/p)$: the ratio $\varphi(x)/x$ depends only on the set of prime divisors of $x$.

Formalization Note Euler's function is Nat.totient. Each equation $\varphi(x)=x/c$ is written without division as $c\cdot\varphi(x)=x$, which is equivalent for natural numbers; the solution sets are stated as equalities of subsets of $\mathbb N$, restricted to $x>0$.