5
$\begingroup$

I was thinking of how to prove $\frac{n^n}{n!}$ is never an integer for $n > 2$. I think if I prove the above question, then this follows immediately.

$\endgroup$
  • 9
    $\begingroup$ Since $\gcd(n,n-1)=1$, no prime factor of $n-1$ divides $n$. And since $n>2$, there must exist a prime dividing $n-1$. $\endgroup$ – Levent Apr 22 '16 at 20:28
  • $\begingroup$ @Levent, brilliant answer... $\endgroup$ – L.F. Cavenaghi Apr 22 '16 at 20:30
  • $\begingroup$ @Levent Great! Should have thought of that! If you write an answer, I will accept it. $\endgroup$ – taninamdar Apr 22 '16 at 20:33
16
$\begingroup$

Since $\gcd(n,n-1)=1$, no prime factor of $n-1$ divides $n$. And since $n>2$, there must exist a prime dividing $n-1$.

$\endgroup$

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.