Solving the congruence $9x \equiv 3 \pmod{47}$ For this question $9x \equiv 3 \pmod{47}$.
I used euler algorithm and found that the inverse is $21$ as $21b-4a=1$
when $a=47$ and $b=9$
I subbed back into the given equation:
\begin{align*}
(9)(21) & \equiv 3 \pmod{47}\\
189 & \equiv 3 \pmod{47}\\
63 & \equiv 1 \pmod{47}
\end{align*}
and I'm stuck, should I divide $63$ by $9$ to get $7$? but it does not comply to the given equation as when $x=7$, it would become $16 \pmod{47}$. 

Using the Extended Euclidean Algorithm as implemented in this answer,
$$
\begin{array}{r}
&&5&4&2\\\hline
1&0&1&-4&9\\
0&1&-5&21&-47\\
47&9&2&1&0\\
\end{array}
$$
shows that $21\cdot9-4\cdot47=1$ so we want $63\cdot9-12\cdot47=3$ which can be reduced, by subtracting $47$ from $63$ and $9$ from $12$, to
$$
16\cdot9-3\cdot47=3
$$
