Introductory algebra course, having trouble proving this one. For monoid $(\mathbb{Z}_n, *)$ defined by multiplication mod n, for each $a \in \mathbb{Z}_n$, define $f:\mathbb{Z}_n \rightarrow \mathbb{Z}_n$ as $f_a(x) = ax$. Prove $f_a(x)$ is a homomorphism.
I'm having trouble because if it's a homomorphism, then $\forall x, y \in \mathbb{Z}_n$, $f_a(xy)=f_a(x)*f_a(y) \Rightarrow a(xy) = axay$
That is where I'm pretty sure I'm not understanding something. That is clearly not true in general, right? Unless I did the calculations wrong, taking $a=2, n=25, x=1, y=7$ for example yields a counterexample where the the equation is not equal.
Thank you in advance