I'm trying to follow a step in a proof, which involves finding $n\in\mathbb{Z}$ such that $\frac{(n^2+3)(n^2-5)}{16n}\in\mathbb{Z}$.
The proof then states that
$\text{hcf}(n,n^2+3)$ divides 3, and
$\text{hcf}(n,n^2-5)$ divides 5.
Hence n divides 15.
I can see 1. and 2. hold, as $\text{hcf}(b,a+mb)=\text{hcf}(a,b)$ But I'm not exactly sure what argument to use to deduce 3.
Any hints would be appreciated. Thanks :)