<s>[INST] Mathematical Induction and Divisibility $7\mid 11^n-4^n$ Prove $11^{n} - 4^{n}$ is divisible by 7 for all natural numbers n.
I know this holds for the base case of n = 1. Am I supposed to use $11^{n + 1} - 4^{n + 1}$ for the induction step?
Any help would be greatly appreciated!
Thank you in advance!
 [/INST] Another way to to do it without induction is to note that the mod cycles for powers of $11$ and $4$ modulo congruent to $7$ correspond to each other. 
$$\begin{equation}\begin{aligned}11 &\equiv 4 \pmod 7\\ 11^2 &\equiv 2 \pmod 7\\ 11^3 &\equiv 1 \pmod 7 \\ 11^4 &\equiv 4 \pmod 7 \\ &\vdots\end{aligned}\end{equation}$$
$$\begin{equation}\begin{aligned}4 &\equiv 4 \pmod 7\\ 4^2 &\equiv 2 \pmod 7\\ 4^3 &\equiv 1 \pmod 7 \\ 4^4 &\equiv 4 \pmod 7 \\ &\vdots\end{aligned}\end{equation}$$
So for any $n$, $11^n \equiv 4^n \equiv r \pmod 7$, so $11^n - 4^n \equiv r - r \pmod 7 \implies 11^n - 4^n \equiv 0 \pmod 7$ as desired. 
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