If I Know that $d\mid n+5$ then How Do I Know that $d\mid(n+5)(n-5)$

Question

Furthermore,

How may I show that the gcd of two expressions is coprime (equal to 1) e.g. $n+3$ and $n^2+3$ given that $n$ is a multiple of 6.

I tried rearranging these expressions and showing that $d|n+3$ and $d|n^2 + 3$, and I then concluded that $d|6$; but how does this necessarily show that $n+3$ and $n^2 + 3$ are coprime?

How can I word it and explain it in a proof?

Thanks

Answer 1

Let $n+5=kd$, where $k\in\mathbb Z$.

Thus, $$\frac{(n+5)(n-5)}{d}=\frac{kd(n-5)}{d}=k(n-5)\in\mathbb Z.$$

Answer 2

If $d$ is a factor of $n+5$ then it must also be a factor of any multiple of $n+5$, such as $(n+5)(n-5)$.