The product of three consecutive natural numbers is divisible by 

Let $n,n+1,n+2$ be there consecutive natural number and let $P\left(n\right)$ be their product. Then,

$P\left(n\right)=n(n+1)(n+2)$

We have

$P\left(1\right)=1\times 2\times 3=6$ which is divisible by 3 and 6

$P\left(2\right)=2\times 3\times 4=24,$ which is divisible by 3, 8 and  6

$P\left(3\right)=3\times 4\times 5=60$ which is divisible by 3 and 6.

$P\left(4\right)=4\times 5\times 6=120$ which is divisible by 3, 8 and 6.

Hence, $P\left(n\right)$is divisible by 6 for all $n \in N$