TITLE: Prove $3|n(n+1)(n+2)$ by induction
QUESTION [6 upvotes]: I tried proving inductively but I didn't really go anywhere. So I tried:

Let $3|n(n+1)(n+2)$.
Then $3|n^3 + 3n^2 + 2n \Longrightarrow  3|(n(n(n+3)) + 2)$

But then?

REPLY [2 votes]: The binomial coefficients are all integers (various proofs). Hence $$\binom r3=\frac {r(r-1)(r-2)}{6}$$ is an integer. Let $r=n+2$, then $6$ is a factor of your product and $3|6$.
In a similar way you can prove that the product of $k$ consecutive integers is divisible by $k!$ - this cam also be done by induction (which is hidden in the statement "the binomial coefficients are all integers" and implicit in the construction of Pascal's Triangle.