Example 10 Find the least common multiple of 513, 135, and 3114.
Since
$1\left|\begin{array}{r|r}513 & 135 \\ 405 & 108 \\ \hline 108 & 27 \\ 108 & \\ \hline 0 & 27\end{array}\right|^{3}$

Therefore, $(513,135)=27$, by Lemma 10 we have
$$\{513,135\}=\frac{513 \times 135}{27}=2565 .$$

Since
\begin{tabular}{|c|c|c|}
\hline 1 & 2565 & \multirow{2}{*}{\begin{tabular}{l}
3114 \\
2565
\end{tabular}} \\
\hline & 2196 & \\
\hline 1 & 369 & 549 \\
\hline & 360 & 369 \\
\hline 20 & 9 & 180 \\
\hline & & 180 \\
\hline & & 0 \\
\hline
\end{tabular}

Therefore, $(2565,3114)=9$, by Lemma 10 we have
$$\{2565,3114\}=\frac{2565 \times 3114}{9}=887490$$

Therefore,
$$\{513,135,3114\}=887490 .$$