Show that $\Phi_n(X)$ has integer coefficients.
The proofs here states that $$\Phi_n(X)=\frac{X^n-1}{\prod_{d|n,d\ne n}\Phi_d(X)}.$$
And by long division, they get $\Phi_n(X)\in \Bbb{Q}[X]$. However, I am unable to understand how $\Phi_n(X)\in \Bbb{Q}[X]$. After that, I can follow. The proof follows by using Gauss's lemma and since $\Phi_n(X)\in \Bbb{Q}[X]$ and $ \prod_{d|n,d\ne n}\Phi_d(X),~~X^n-1\in \Bbb{Z}[X]\implies \Phi_n(X)\in\Bbb{Z}[X]$.
In general:
If we have $g(x)=\frac{p(x)}{h(x)}$ where $p(x)\in\Bbb{Z}[X], h(x)\in\Bbb{Z}[X]$, prove that $g(x)\in\Bbb{Z}[X]$.
For this, if we can show that $g(x)\in \Bbb{Q}[X]$, then the proof is just same as above.