TITLE: How to prove that $(10n+2)!$ is divisible by $(10n+4)$
QUESTION [8 upvotes]: I would like to know how one can prove that $(10n+2)!$ is divisible by $(10n+4)$ for natural $n > 0$. I tried to do this using induction but I've got stuck because I couldn't simplify the expression. 
I first proved that it holds for $n = 1$.
$$\frac{(10\cdot1+2)!}{(10\cdot1+4)}=\frac{12!}{14}=34214400$$
I then assumed that it's true for n.
$$(10\cdot n+2)!=k(10\cdot4)$$ where k is a natural number.
I then tried to prove that this is true for n+1 using the previous assumption. 
I got stuck here because I couldn't prove that $(10n+12)!$ is divisible by $(10n+14)$.
I would very much appreciate your help , thanks.

REPLY [26 votes]: Hint $\ 10n\!+\!4\, =\, 2(5n\!+\!2)\ $ and $\,2\neq 5n\!+\!2\,$ are both $< 10n\!+\!2\ $ for $\ n> 0\,$ so occur in $(10n\!+\!2)!$