<s>[INST] Proving that $3^n+7^n+2$ is divisible by 12 for all $n\in\mathbb{N}$. Can someone help me prove this? :( I have tried it multiple times but still cannot get to the answer.
Prove by mathematical induction for $n$ an element of all positive integers that   $3^n+7^n+2$ is divisible by 12.

 [/INST] With induction:
When $n=1$: $3^1+7^1+2=12$ is divisible by $12$. Therefore, we assume that $(3^k+7^k+2)$ is divisible by $12$ for some $k≥1$ (thus, $3^k+7^k+2=12a$ where $a$ is a function of $3$ and $7$).
Then, $$3^{k+1}+7^{k+1}+2=3×3^k+7×7^k+2$$
$$=3×3^k+7^k(3+4)+6-6+2$$
$$=3×3^k+3×7^k+6+4×7^k-4$$
$$=3(3^k+7^k+2)+4(7^k-1)$$
$$=3×(12a)+4(7-1)(7^{k-1}+7^{k-2}+⋯+2+1)$$
$$=36a+24(7^{k-1}+7^{k-2}+⋯+2+1)$$
$$=12b$$ where $b$ is also a function of $3$ and $7$.
Thus, $(3^{k+1}+7^{k+1}+2)$ is also divisible by $12$.
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