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 $30$.

You have to prove that $x-x^9$ is divisible by $2$, $3$ and $5$.


*

*$x\equiv x^9\pmod{2}$ is obvious, isn't it?

*By Fermat's little theorem, $x^3\equiv x\pmod{3}$, so $x^9=(x^3)^3\equiv x^3\equiv x\pmod{3}$

*By Fermat's little theorem, $x^5\equiv x\pmod{5}$, so $x^9=x^4x^5\equiv x^4 x\equiv x^5\equiv x\pmod{5}$.
