Proof Checking and input: Generators of $\mathbb{Z}_{pq}$ I'm self-studying abstract algebra (slowly but surely), and I have a question about my answer to the following prompt:
Problem statement:

Show that there are $(q-1)(p-1)$ generators of the group $\mathbb{Z}_{pq}$, where $p$ and $q$ are distinct primes.
  (Where $\mathbb{Z}_{n}$ is the additive group of integers modulo $n$)

I can use a theorem from my textbook that says:

The integer $r$ generates the group $\mathbb{Z}_n$ iff
  $$1\le r\lt n \quad\text{and}\quad \gcd(r, n) = 1$$

My attempt at a proof:
Let $p$ and $q$ be distinct primes and $\mathbb{Z}_{pq}$ be the additive group of integers modulo $pq$.
An element $a \in \mathbb{Z}_{pq}$ is a generator of $\mathbb{Z}_{pq}$ iff:
$$1\le a\lt pq \quad\text{and}\quad \gcd(a, pq) = 1$$
There are $p-1$ positive multiples of $q$ less than $pq$.  Also, there are $q-1$ positive multiples of $p$ less than $pq$.  These are the only elements of $\mathbb{Z}_{pq}$ that are not coprime to $pq$.  Finally, $0$ is not a generator of $\mathbb{Z}_{pq}$.
There are $pq$ elements in $\mathbb{Z}_{pq}$, so the number of generators is:
$$\begin{align}
pq - (p-1) - (q-1) - 1 &= pq -p -q + 1 \\
&= p(q-1) - (q - 1) \\
&= (q-1)(p-1)
\end{align}$$
My questions:


*

*Obviously, if there's a flaw in the proof, I'd like to know. :)  Aside from that...

*When I assert "these are the only elements of $\mathbb{Z}_{pq}$ that are not coprime to $pq$," do I need to further show that this is true?  It is patently obvious to me, but I know that just claiming "this is obvious" is not a valid method of proof.

*I'd like to have some input on the style/format of my proof.  Is there a better (read: more formal/traditional) way to phrase something in this proof, or is there a format that I'm not following?  As I'm self-teaching, I don't want to learn bad habits...
 A: I don't see anything wrong with your proof.
For your second question, you could expand a bit on that statement, perhaps saying something like "since $1$,$p$, $q$ and $pq$ are the only divisors $pq$, any integer $n$not divisible by $p$ or $a$ must have $\gcd(n,pq)=1$, and be coprime to $pq$", but in situations like this, where a moment's though and writing down a few definitions will give a proof, omitting it is usually safe.
In writing proofs, I have found the following idea helpful: a proof is really intended to convince someone (a reader, a teacher, yourself) that a theorem is true. If, after reading your proof and spending a little time thinking about each step, this person could still doubt that your theorem is true, then you should add more. Otherwise, your safe. This reasoning, however, is very audience-dependant. For some audiences, the following would be an acceptable proof of your theorem:

The generators of $\Bbb{Z}_{pq}$ correspond with integers less than and coprime to $pq$. Since these are counted by Euler's $\phi(n)$, $\phi(n)$ is multiplicative, and, for any prime $p$, $\phi(p) = p-1$, the number of generators of $\Bbb{Z}_{pq}$ is $\phi(pq)=\phi(p)\phi(q) = (p-1)(q-1)$.

For lots of other audiences, it would not be.
Remember, a proof doesn't always have to be written down, finalized, and perfect. It can be more of a conversation.
