Let $p$ be a prime number, and $x, n$ integers with $n \geqslant 1$.

(i) Prove Fermat's Little Theorem: for any integer $x, x^{p} \equiv x(\bmod p)$.

(ii) Show that if $y$ is an integer such that $x \equiv y\left(\bmod p^{n}\right)$, then for every integer $r \geqslant 0$,

$$x^{p^{r}} \equiv y^{p^{r}}\left(\bmod p^{n+r}\right)$$

Deduce that $x^{p^{n}} \equiv x^{p^{n-1}}\left(\bmod p^{n}\right) .$

(iii) Show that there exists a unique integer $y \in\left\{0,1, \ldots, p^{n}-1\right\}$ such that

$$y \equiv x \quad(\bmod p) \quad \text { and } \quad y^{p} \equiv y \quad\left(\bmod p^{n}\right)$$