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To prove $10^n \equiv (-1)^n\pmod{11}$, $n\geq 0$, I started an induction.

It's $$11|((-1)^n - 10^n) \Longrightarrow (-1)^n -10^n = k*11,\quad k \in \mathbb{Z}. $$

For $n = 0$: $$ (-1)^0 - (10)^0 = 0*11 $$

$n\Rightarrow n+1$ $$\begin{align*} (-1) ^{n+1} - (10) ^{n+1} &= k*11\\ (-1)*(-1)^n - 10*(10)^n &= k*11 \end{align*}$$ But I don't get the next step.

12 Answers 12