What is the set of units in $ (\mathbb Z/n \mathbb Z)$ equal to $\{k+n \mathbb Z : gcd(n,k)=1\} $

Question

I have been learning about quotient rings, and have come across the following ring: $ (\mathbb Z/n \mathbb Z)$, the ring of residue classes mod $n$.

I have found that the set of units in $ (\mathbb Z/n \mathbb Z)$ is $$ (\mathbb Z/n \mathbb Z) ^X = \{k+n \mathbb Z : gcd(n,k)=1\} $$

How can I see that this is true?

Answer 1

One inclusion is straightforward. The other follows from the existence of the extended Euclidean algorithm.