Zero divisors of ${\Bbb Z}_n = $ integers $\!\bmod n$ Consider the following proposition:

A nonzero element $m\in{\bf Z}_n$ is a zero divisor if and only if $m$ and $n$ are not relatively prime.

I don't know if this is a classical textbook result. (I didn't find it in Gallian's book). 
For the "only if" part, one may like to use the Euclid's lemma. But I cannot see how can one prove the "if" part:
If $m_1>0$, $(m_1,n)=d>1$, and $n|m_1m_2$, then $n\nmid m_2$. 
Edit:
The "if" part, should be:
If $m_1>0$ and $(m_1,n)=d>1$, then there exists $m_2$ such that $n|m_1m_2$, and $n\nmid m_2$.
Does one need any other techniques other than "divisibility"?
Questions:


*

*How to prove the proposition above?  

*How many different proofs can one have?


Hint $\rm\,\ d\mid n,m\,\Rightarrow\,\ mod\ n\!:$ $\rm\displaystyle\:\ 0\equiv n\:\frac{m}d\ =\ \frac{n}d\: m\ $ and $\rm\, \dfrac{n}d\not\equiv 0\,$ if $\rm\,d>1$
