<s>[INST] Prove by induction that $3^n +7^n −2$ is divisible by $8$ for all positive integers $n$... Prove by induction that $3^n +7^n −2$ is divisible by $8$ for all positive integers $n$.
So far I have the base case completed, and believe I am close to completing the proof itself.
Base case:$(n=1)$
$3^1 + 7^1 - 2 = 8/8 = 1 $
Inductive Hypothesis: Assume that $3^n +7^n −2$ is divisible by 8 for all positive integers n.
Induction step $(n+1)$ case:
$$ 3^{n+1} + 7^{n+1} - 2 $$
$$3(3^{n}) + 7(7^{n}) - 2$$
$$3^n + 7^n = 8x $$
-It  seems to me that this could be the end of the proof because whatever the answer is would be a multiple of 8: but I am unsure, any help is appreciated. 
 [/INST] We have that $3^n+7^n-2=8k$ for some $k$. Now we substitute this expression into $3(3^n)+7(7^n)-2$ to get:

$3(8k-7^n+2)+7(7^n)-2=24k+4(7^n)+4=24k+4(7^n+1)$ where $7^n+1$ is even so we may rewrite $24k+4(7^n+1)=24k+4(2m)=24k+8m$ which is divisible by $8$.
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