Prove by induction that $n(n+1)(n+5)$ is multiple of $3.$ $n(n+1)(n+5) = 3d$
I cannot figure out how to solve this homework question. A friend gave me a solution I couldn't make sense of, and I hope there's something easier out there. Also, what would be the general approach towards questions of this form?

The proof can be done without induction, just see that $n$ is of one of the three forms 
$n=3k$, $n=3k+1$ or $n=3k+2$. and check each of these cases
Suppose  $n=3k$ and $n(n+1)(n+5)=3k(3k+1)(3k+5)=3(k(3k+1)(3k+5))=3d$  where $d=k(3k+1)(3k+5)$, 
Next suppose that $n=3k+1$, hence
$n(n+1)(n+5)=(3k+1)(3k+2)(3k+6)=3((3k+1)(3k+2)(k+2))=3d$  where $d=(3k+1)(3k+2)(k+2)$,
Finally suppose that $n=3k+2$, hence
$n(n+1)(n+5)=(3k+2)(3k+3)(3k+7)=3((3k+2)(k+1)(3k+7))=3d$  where $d=(3k+2)(k+1)(3k+7)$.
Since it works in all three cases it must be true for any $n$
