How to prove that $(10n+2)!$ is divisible by $(10n+4)$

Question

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.

Answer 1

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