Prove that $n^3+2$ is not divisible by $9$ for any integer $n$ How to prove that $n^3+2$ is not divisible by $9$?

Suppose, $\exists n\in \mathbb{N}$ such that $n^3+2\equiv 0 \pmod{9}\implies n^{6}\equiv 4\pmod{9}$, which is not true since\begin{array}{rl}
n^6\equiv 1\pmod 9 & \mbox{if $n$ and $9$ are relativley prime,  by Euler's theorem, since $\phi(9)=6$.}\\ 
n^6\equiv 0\pmod 9 & \mbox{otherwise} 
\end{array}  
