What is the easiest ways to prove this: $3^n-1$ is divisible by $4 \implies n $ is even? Moreover, how would I figure out that $n$ must be even if I didn't know the result?

My approach is this: suppose $n=2k+1$. Consider $3^{2k+1}-3$. If we show that it is divisible by $4$, it will follow that $3^n-1$ has remainder $2$ in the division by $4$ if $n$ is odd. We have $3^{2k+1}-3=3(3^k-1)(3^k+1)$. Each of the two brackets is divisible by $2$, so the whole thing is divisible by $4$. Is this correct?