Is it true for $n > 2$ then there always exists a prime $\le n$ that does not divide $n$?

Question

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.

Answer 1

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