Equivalence Classes Problem

Question

Let $R$ be the relation on $\mathbb{N} \setminus \{ 0 \}$ given by $R = \{ (x,y) \in (\mathbb{N} \setminus \{ 0 \}) \times (\mathbb{N} \setminus \{ 0 \}):$ there exists $k \in \mathbb{N}$ such that $x| y^k$ and $y| x^k \}.$ Describe in detail the equivalence classes.

Answer 1

Hint: Consider the function $f:\Bbb N\setminus\{0\} \to \mathcal P(\Bbb N), \ n\mapsto \{p\in\Bbb N:p$ is a prime divisor of $n\}$,
and prove that its kernel is $R$, i.e. $(x,y)\in R\iff f(x)=f(y)$.