Cyclotomic polynomial identity

Question

Let $n,m$ be positive integers and $\zeta_{n}=e^{\frac{2\pi i}{n}}$.Define the polinomial

$$\phi_{n,m}(t)=\prod_{(j,n)=1}(t-\zeta_{n}^{mj})$$

let $l=(n,m)$. I must to prove that

$$\prod_{d|m}\phi_{n,d}(t)=(t^{n/l}-1)^{l}$$

How can I prove it?