TITLE: Proof By Induction Divisibility Question: $12\mid 3^n + 7^{n-1} + 8$
QUESTION [5 upvotes]: Prove that $3^n + 7^{n-1} + 8$ is divisible by $12$ for all positive integers $n$.
I have proved it is true for $n=1$ and I have done the 'assume $n=k$' step, but after getting $3^{k+1} + 7^k + 8$, I am stuck. Where do I go from here?

REPLY [5 votes]: HINT: $3^{k+1}+7^{k}+8=7(3^{k}+7^{k-1}+8)-4\cdot3^{k}-6\cdot8=7(3^{k}+7^{k-1}+8)-12(3^{k-1}+4)$
Clarification of where this comes from:
$3^{k+1}=3\cdot3^{k}=7\cdot3^{k}-4\cdot3^k$
similarly $8=7\cdot8-6\cdot8$