[Math] Prove that $7^n+2$ is divisible by $3$ for all $n ∈ \mathbb{N}$

Question

Use mathematical induction to prove that $7^{n} +2$
is divisible by $3$ for all $n ∈ \mathbb{N}$.

I've tried to do it as follow.

If $n = 1$ then $9/3 = 3$.
Assume it is true when $n = p$. Therefore $7^{p} +2= 3k $ where $k ∈ \mathbb{N} $. Consider now $n=p+1$. Then
\begin{align}
&7^{p+1} +2=\\
&7^p\cdot7+ 2=\\
\end{align}
I reached a dead end from here. If someone could help me in the direction of the next step it would be really helpful. Thanks in advance.

Answer 1

Hint: $(7^{p+1}+2)-(7^p+2)=6\times7^p$.