elementary number theory - Prove that $9mid (4^n+15n-1)$ for all $ninmathbb N$

First of all I would like to thank you for all the help you've given me so far.

Once again, I'm having some issues with a typical exam problem about divisibility. The problem says that:

Prove that $\forall n \in \mathbb{N}, \ 9\mid4^n + 15n -1$

I've tried using induction, but that didn’t work. I've tried saying that:

$4^n + 15n-1 \equiv 0 \pmod{9}$. Therefore, I want to prove that $4^{n+1} + 15(n+1) -1 \equiv 0 \pmod{9}$.

I've prooved for $n=1$, it's $18\equiv 0 \pmod{9}$, which is OK.

But for the inductive step, I get:

$4\cdot4^n + 15n+15-1 \equiv 0 \pmod{9}$

And from there, I don't know where to replace my inductive hypothesis, and therefore, that's why I think induction is not the correct tool to use here. I guess I might use some tools of congruence or divisibility, but I'm not sure which ones.

I do realize that all $n\in \mathbb{N}/ \ 3 \ |\ n \Rightarrow 4^n \equiv 1 \pmod{9} \text{ and } 15n \equiv 0 \pmod{9}$. In that case, where 3 divides n, then I have prove that $4^n + 15n-1 \equiv 0 \pmod{9}$. But I don't know what to do with other natural numbers that are not divisible by 4, that is, all $n \in \mathbb{N} / n \equiv 1 \pmod{3} \text{ or } n \equiv 2 \pmod{3}$.

Any ideas? Thanks in advance!

Answer

By the Inductive Hypothesis, $4^n + 15n -1 \equiv 0$ so $4^n \equiv 1-15n$ and thus
$$4^{n+1}+15(n+1)-1 = 4 \cdot 4^n + 15n + 14 \equiv 4 \cdot (1-15n) + 15n + 14 = 18 -45n \equiv 0$$
since both $18$ and $45$ are divisible by 9.