<s>[INST] Prove that $6^n+8^n$ is divisible by $7$ iff $n$ is odd The question:

Prove that $6^n+8^n$ is divisible by $7$ iff $n$ is odd.

Hence, prove $7 | 6^n + 8^n \iff n ~$ is odd
I realise that this is a proof by induction, and this is what I have so far:
\begin{align}
f(n) & = 6^n + 8^n \\
& = (2\cdot 3)^n + (2^3)^n \\
& = 2^n \cdot 3^n + 2^{3n} \\
\end{align}
\begin{align}
\text{Assume} ~ f(k) & = 6^k + 8^k \\
& = 2^k \cdot 3^k + 2^{3k} \\
f(k+1) & = 2^{k+1} \cdot 3^{k+1} + 2^{3(k+1)} \\
\end{align}
Where do I go from here?
 [/INST] Because $$6^n+8^n=(6+8)(6^{n-1}-6^{n-2}\cdot8+...-6\cdot8^{n-2}+8^{n-1}).$$
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