# proving a modular arithmetic and using the fact to compute

So i was given this question

Show that for any natural n

$10^n(mod11)$ $=$ $\left\{ \begin{array}{lcc} 1, & \mbox{ n even}\\ \\ 10, &\mbox{ n odd} \\ \\ \end{array} \right.$

Using this fact compute $654321 (mod 11)$

My problem is im used to just solving the modular arithmetic not proving. Like i know $654321 (mod 11)$ = $8$, but how would i prove the fact and use it?

$$10 \equiv 10 \mod{11}$$ $$10^2 \equiv 10 \cdot 10 \equiv 1 \mod{11}$$ $$10^3 \equiv 1 \cdot 10 \equiv 10 \mod{11}$$ $$\vdots$$
$10^n \equiv (-1)^n \pmod{11}$.