If $\gcd (a,n) = 1$, how can we prove that $\gcd(a + kn, n) = 1$?

Question

If $\gcd (a,n) = 1$, then how can we prove that $\gcd(a + kn, n) = 1$, where $k$ is any integer?

I feel like I should start with linear combinations theorem, but I'm quite lost. Any help would be greatly appreciated!

Answer 1

If some prime $p|n$, $p|a+kn$, then $p|kn$ and...