Why does $n \mid (x-1)\iff x=kn+1$?

Question

It maybe trivial but I can convince myself, why is the following true?

$$n \mid (x-1)\iff x=kn+1$$

Or in the general case

$$n \mid (x-a)\iff x=kn+a:a\in \mathbb{Z}$$

Answer 1

By definition $n\mid(x-a)$ means there exists $k\in\Bbb{Z}$ such that $x-a=kn$. This is equivalent to $x=kn+a$.

Answer 2

It is true, because by definition $n\mid (x-1)$ means that there exists a $k$ such that $kn=x-1$. The definition $n\mid m$ says that there is a $k$ such that $kn=m$. So the "general case" is just the definition, for different $m$'s.