How can I solve $\frac{xy}{3} =3$ in $\mathbb{Z}_{5}$?

Question

How can I solve $\frac{xy}{3}=3$ in $\mathbb{Z}_{5}$?

I think about that this way $xy=9 \rightarrow xy=4$. It seems that $x=1$ and $y=4$ is a solution, but than I get

$$\frac{1\cdot 4}{3}=3$$

What is my problem?

Answer 1

There is no problem. Observe that in $\mathbb{Z}_{5}$ one has

$$3\cdot 2\equiv 6\equiv 1\mod{5}$$

so

$$\frac{1}{3}\equiv 2\mod{5}$$

and indeed

$$\frac{4}{3}\equiv 4\cdot 2\equiv 8\equiv 3\mod{5}$$

exactly what you've got!