Example 3. Find the integer solutions of $36 x+83 y=1$.

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$$
\begin{array}{l}
\text{Since } (36,83)=1, \text{ by Theorem 1, there must be two integers } x, y \text{ that are solutions to the equation.} \\
\because 83=2 \times 36+11, 36=3 \times 11+3, \\
11=3 \times 3+2, \quad 3=2+1, \\
\therefore 1=3-2=3-(11-3 \times 3)=4 \times 3-11 \\
=4 \times(36-3 \times 11)-11 \\
=4 \times 36-4 \times 3 \times 11-11 \\
=4 \times 36-11 \times(12+1) \\
=4 \times 36-13 \times 11 \\
=4 \times 36-13 \times(83-2 \times 36) \\
=4 \times 36-13 \times 83-13 \times(2 \times 36) \\
=36(4+26)-13 \times 83 \\
=36 \times 30-83 \times 13. \\
\end{array}
$$

Therefore, \( x=30, y=-13 \)