If $a\in\mathbb{Z}, n\in\mathbb{N}$, then the equation $xa\equiv1\pmod {n}$ has a solution for some $x\in\mathbb{Z}$.
I'm not quite sure where to start. I know that $n|(xa-1)$, so $ns=xa-1$ for some integer $s$.
Should I start plugging in numbers to find one that makes it possible for a to divide $(ns+1)$?
$(xa-1)\equiv0 \pmod {n}$, so this means that $xa=\pm 1$?
I feel so dumb when it comes to proofs, so please go easy on me.
Thank you.