I'm struggling to find a beginning for finding an equivalence class involving lcm. Given a fixed $n \in \mathbb{N}$ and relation $R = \{(a,b) \in \mathbb{Z}^2 \: | \: lcm(a,n) = lcm(b,n) \}$, find the equivalence class of $p$ where $p$ is prime.
I don't have much experience dealing with lcm's in regards to equivalence relations. I understand the ideas between them separately but have a bit of trouble figuring how they relate in this case. How does $p$ relate to $R$?
If p|n, then lcm(p,n) = n and [p] = { q : q|n }.
If not p|n, then lcm(p,n) = pn and [p] = { pq : q|n }.