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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?

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$$10 \equiv 10 \mod{11}$$ $$10^2 \equiv 10 \cdot 10 \equiv 1 \mod{11}$$ $$10^3 \equiv 1 \cdot 10 \equiv 10 \mod{11}$$ $$\vdots$$

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$10^n \equiv (-1)^n \pmod{11}$.

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