Let $R$ be the relation in $A:=\{1,2,3,...,23,24,25\}$ defined as $nRm$ $\iff$ $nm$ is a square number in $\mathbb N$. Prove that $R$ is an equivalence relation and determine the partition in $A$ defined by the relation $R$.

I am having problems with two parts of the exercise: 1) proving transitivity and determining the partition. So here's what I've done so far:

If a relation $R$ is reflexive, symmetric and transitive then $R$ is an equivalence relation. So I have to prove that $R$ is reflexive, symmetric and transitive.

Reflexivity: Let $n \in \mathbb A$, $nRn \iff nn$ is a square natural number. But $nn=n^2$ which is a square number and is natural by closure of multiplication in $\mathbb N$.

Symmetry: Let $n,m$ $\in A$. Suppose $nRm \implies nm$ is a square natural number. But $nm=mn$, which means $mn$ is a square natural number $\implies$ $mRn$.

Transitivity: Let $n,m,p \in A$ and suppose $nRm$ and $mRp$, this means $nm=s^2$ and $mp=t^2$. Then $m \vert s^2$ and $m\vert t^2$ so $np=\frac{s^2t^2} {m^2} \in \mathbb N$. I couldn't prove that $np$ is a square number.

For the last part of the exercise, I could figure out the partition but the problem is that I don't know how to formally justify my result. For $1$,$1Rn \implies 1n=n$ is a natural square number. By this condition it's immediate that $\overline 1=\{1,4,9,16,25\}$. For $2$, $2Rn \implies 2n$ is a square natural number. Well, I could see that $\overline 2=\{2,,8,18\}$. I don't know how to put it in symbols that this has to be the equivalent class $2$. Similarly, I had problems to justify all the other equivalent set classes. For example, $6=\{6,24\}$ and for any $p \in A$ with p a prime greater than $5$, I got that $\overline p=\{p\}$. Could someone help me to explain it correctly and in mathematical language?