Question

If n is an integer. Prove that $n(n+1)(n+5)$ is a multiple of 6.

Answer 1

$n(n+1)(n+5)=n(n+1)\left[\right(n+2)+3]$
$=n(n+1)(n+2)+3n(n+1)$
Now, $n(n+1)(n+2)$ being the product of three consecutive integers is divisible by $3!=6$.
$n(n+1)$ being the product of two consecutive integers is divisible by $2!=2$
$\therefore 3n(n+1)$ is divisible by 6
$\therefore n(n+1)(n+5)$
$=n(n+1)(n+2)+3n(n+1)$ is divisible by 6
i.e., $n(n+1)(n+5)=M\left(6\right)$