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.


1 Answer 1


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

  • 1
    $\begingroup$ Thank you. Now I understand. $\endgroup$ Mar 5, 2017 at 22:15

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.