Claim: $$\prod_{d \mid n} \Phi_{d}(x) = x^n - 1 $$ Where $\Phi_{m}(x)$ is the $m^{th}$ cyclotomic polynomial of $x$.
I think it has to do with Euler's Totient Function $\phi$ and the result $$\sum_{d \mid n} \phi(d) = n$$ but am not sure how to show the intermediate step(s).
Edit:
Ok, I would now like to take this a little further and show, by induction on $m$ and use of the above result, that $$\Phi_m (x)\in \mathbb{Z}[x]$$ By basic computation I know it's true for $m<4$. I've considered dividing the products from the initial claim for $n=k$ and $n=k+1$. Do you have any advice?