Prove transitivity of relation $R=\{(a,b) \in Z \times Z | \exists m,n \in N\setminus\{0\}: a^m = b^n \}$ Transitivity means: $(a,b) \in R \quad \land \quad (b,c) \in R \quad \implies (a,c) \in R$ But I'm not really sure how to go about it. I know that there is some $s,t,v,u$ such that $a^s = b^t$ and $b^v = c^u$ but to prove transitivity of the relation I'd have to find some $x,y$ such that $a^x = c^y$. Anyone have a pointer?

HINT: If $a^s=b^t$, then $a^{sv}=(a^s)^v=(b^t)^v=b^{tv}=(b^v)^t=\ldots$