Example 9 Find the greatest common divisor of $198,252,924$, and express it as an integer linear combination of 198,252 and 924.
From Theorem 4(i) of §4 and Example 6 of §2, we know

From this and Example 6 of §2, we get: $(198,252,924)=6$,
$$\begin{aligned}
6 & =924-51 \cdot 18=924-51(4 \cdot 252-5 \cdot 198) \\
& =924-204 \cdot 252+255 \cdot 198 .
\end{aligned}$$
$$\begin{array}{l}
(198,252,924)=((198,252), 924)=(18,924) \text {. } \\
\begin{array}{l|l}
924=51 \cdot 18+6 & 6=924-51 \cdot 18
\end{array} \\
18=3 \cdot 6
\end{array}$$