Since $(36,83)=1$, by Lemma $1-$ there must exist two integers $x$, $y$ such that
$$36 x+83 y=1$$

holds, find $x, y$.
Since
$$\begin{array}{l}
93=2 \times 36+11, \quad 36=3 \times 11+3 \\
11=3 \times 3+2, \quad 3=2+1
\end{array}$$

we get
$$\begin{aligned}
1 & =3-2=3-(11-3 \times 3)=4 \times 3-11 \\
& =4 \times(36-3 \times 11)-11=4 \times 36-13 \times 11 \\
& =4 \times 36-13 \times(93-2 \times 36) \\
& =30 \times 36-13 \times 93 \\
& =36 \times 30-93 \times 13
\end{aligned}$$

Therefore,
$$x=30, \quad y=-13$$