Fix $m\in\mathbf{Z}$ and consider for $x,y\in\mathbf{Z}$ the relation $\sim$, where $x\sim y$ whenever there is a $z\in\mathbf{Z}$ such that $z^2=(x\cdot y)^m$.
Consider for a fixed $n\in\mathbf{N}$ the relation $\Delta$ on $\mathbf{N}$ where $x\Delta y$ whenever there is a $z\in\mathbf{N}$ such that $z^2=(x\cdot y)^n$. I use the convention $0\notin \mathbf{N}$.
How can I show that $\sim$ is not transitive and $\Delta$ is? And how can I determine the cardinality of the equivalence classes under $\Delta$?