Is $n! \nmid n^n$ true for every sufficiently large $n$ (maybe $n \ge 3$)? Is $n! \nmid n^n$ true for every sufficiently large $n$ (maybe $n \ge 3$)? If so, how to prove it? * * * Note that it is easy to show that $n! \nmid n^n$ if $n$ is a prime.

If $n!|n^ n$ then $n-1$ divides $n^n$.

$n$ and $n-1$ are coprime, it follows $n-1$ and $n^n$ are coprime, it follows $n-1$ does not divide $n^n$ unless $n-1=1$ which is not true as $n\geq 3$