Positive Integer Divison Proof Prove that $6|n(n + 1)(n + 2)$ for any integer $n ≥ 1$
I have attempted to do this but never seem to prove it.

Proof by contradiction
Assumption: There are $n$ where $6$ doesn't divide $n(n+1)(n+2)$.   
Let $x$ be the smallest $n$ where $6$ doesn't divide $n(n+1)(n+2)$.   
Since $6|1\times 2\times 3$ and $6|2\times 3\times 4$ that $x$ has to be at least $3$.  
If $6$ doesn't divide $x(x+1)(x+2)$ then it doesn't divide $x(x+1)(x+2)-6x^2 = ... = (x-2)(x-1)x = (x-2)(x-2+1)(x-2+2)$.
So the statement isn't true for $x-2$ and that is a contradiction to the fact that $x$ is smallest $n$ where the statement isn't true.
