How can I demonstrate that $x-x^9$ is divisible by 30? How can I demonstrate that $x-x^9$ is divisible by $30$ whenever $x$ is an integer?
I know that $$x-x^9=x(1-x^8)=x(1-x^4)(1+x^4)=x(1-x^2)(1+x^2)(1+x^4)$$
but I don't know how to demonstrate that this number is divisible by $30$.

It's a special case of the following theorem. The direction we need $(\Leftarrow)$ 
has a very simple proof.
Theorem $\ $  For natural numbers $\rm\:a,e,n\:$ with $\rm\:e,n>1$ 
$\qquad\rm n\:|\:a^e-a\:$ for all $\rm\:a\:\iff  n\:$ is squarefree, and prime $\rm\:p\:|\:n\:\Rightarrow\: p\!-\!1\:|\:e\!-\!1$ 
