For any $(x,y) \in \mathbb{N}$, $xRy $ iff $xy$ is a perfect square. Show that $R$ is an equivalence relation and what are the equivalence classes? Here is my progress so far.
By the rules of multiplication we know that it is already symmetric and reflexive because for any $x \in \mathbb{N}, x^2$ is by definition a perfect square and for any $x,y \in \mathbb{N}, xy = yx$. But for transitivity would I use the fundamental theorem of arithmetic and use the fact that all the exponents over the primes in a perfect square have to be even? Also for the equivalence classes do I do a similar thing and use the prime factorization?
You just need to prove that $x R y$ if and only if $S(x)=S(y)$, where $S(x)$ is the set of prime numbers with odd exponents in the prime factorization of $x$. All the rest follows from that.