Assume that exist $n: n^3 - 9n + 27 \vdots 81 \Rightarrow n \vdots 3$. Rewrite the expression as $(n-3)n(n+3) + 27$. Because $n \vdots 3$, we have $n-3, n+3$ both divisible by $3$. Furthermore, at least one of $n-3, n, n+3$ is divisible by $9$ so $(n-3).n.(n+3) \vdots 81$ but $27 \not \vdots 81 \Rightarrow (n-3)n(n+3) + 27$ is not divisible by $81$ (contradiction)