Solving $217 x \equiv 1 \quad \text{(mod 221)}$ I am given the problem:
Find an integer $x$ between $0$ and $221$ such that
$$217 x \equiv 1 \quad \text{(mod 221)}$$
How do I solve this? Unfortunately I am lost.

Using the Euclid-Wallis Algorithm:
$$
\begin{array}{r}
&&1&54&4\\
\hline
\color{red}{1}&0&1&\color{red}{-54}&217\\
0&\color{green}{1}&-1&\color{green}{55}&-221\\
\color{red}{221}&\color{green}{217}&4&\color{blue}{1}&0
\end{array}
$$
we get that $\color{green}{55\cdot217}\color{red}{-54\cdot221}=\color{blue}{1}$. Thus, $x=55\pmod{221}$.
