Example 6 Find the greatest common divisor of 198 and 252, and express it as an integer linear combination of 198 and 252.
\begin{tabular}{|c|c|c|c|}
\hline $252=1 \cdot 198+54$ & & & $18=-198+4(252-198)$ \\
\hline $198=3 \cdot 54+36$ & & & $=4 \cdot 252-5 \cdot 198$ \\
\hline $54=1 \cdot 36+18$ & $\gamma$ & $\uparrow$ & $18=54-(198-3 \cdot 54)$ \\
\hline $36=2 \cdot 18$ & & & $=-198+4 \cdot 54$ \\
\hline & & & $18=54-36$ \\
\hline
\end{tabular}