Definition of modulo division Could anyone tell me why
$3/7\equiv 9 \mod12$? How is this operation defined and why sometimes modulo division is impossible, like in the case of $3/4 \mod 12$?
See here for more details, I'm refering to page 95.
I can't grasp why $3/7\equiv 9 \mod12$. I divide $3$ by $7$, what I get is $3 = 0 \times 7 + 3$, the remainder is $3$. Now how is it equal to $9$ mod $12$?

Think of it like this:
$$\frac{3}{7} = 3 \cdot \frac{1}{7} = 3 \cdot 7^{-1}.$$
So then
\begin{align}
  \frac{3}{7} \mod 12 &= (3 \cdot 7^{-1}) \mod 12\\
            &= (3 \mod 12) \cdot (7^{-1} \mod 12).
\end{align}
$3 \mod 12$ is just 3.  What's $7^{-1} \mod 12$?  In other words, find $x$ so that $7x \equiv 1 \mod 12$.  Then multiply this $x$ by 3 and evaluate modulo 12 again.
