Calculate $\frac{1}{3}$ in $\mathbb{Z}_{5}$.

 untranslated text:
Calculer $\frac{1}{3}$ dans $\mathbb{Z}_{5}$.
We find the Bézout coefficients (using, for example, the extended Euclidean algorithm) to progress from decimal to decimal:

$$
1=3 \times 2-5 \times 1
$$

so

$$
\frac{1}{3}=2+5 \times \frac{-1}{3}
$$

We know that $\frac{-1}{3}$ is a $p$-adic integer, so the first decimal of $\frac{1}{3}$ is 2. Similarly,

$$
\begin{aligned}
& -1=3 \times 3-5 \times 2 \text { so } \frac{-1}{3}=3+5 \times \frac{-2}{3} \\
& -2=3 \times 1-5 \times 1 \text { so } \frac{-2}{3}=1+5 \times \frac{-1}{3}
\end{aligned}
$$

We have already encountered $\frac{-1}{3}$, so we have a periodicity.

We deduce: $\frac{1}{3}=\ldots 13 \ldots 13132$.