TITLE: Proof that if $a$ is coprime to $n$ in $R = \mathbb{Z}/(n)$, then $a$ is a unit
QUESTION [5 upvotes]: Proof that if $a$ is coprime to $n$ in $R = \mathbb{Z}/(n)$, then $a$ is  a unit
Let $a \in R$ be coprime to $n$.
Then $gcd(a, n) = 1$
$\implies a\nmid n$
Let $b \in R$.
Now as $a \nmid n$ we have that $ba \neq 0$
I.e. $ba \in \{1,...,n-1\}$
I.e. ba will eventually 'hit' all numbers in $\{1, 2, ...n-1\}$, and in particular for some value of $b$ we will get that $ba = 1$.

REPLY [2 votes]: Not to answer a question with another question, but:
What about the old saw that $\gcd(a, n) = 1$ implies there exist $c, d \in \Bbb Z$ with $ac + dn =1$ whence, reducing $\mod n$, we have $ac = 1$ in $R$, so $a$ is a unit in $R$?
Hmmm . . . I guess that's the Bezout's identity approach . . . just checked out robjohn's comment.
Hope this helps.  Cheerio,
and as always,
Fiat Lux!!!