Example 11 Find the least common multiple of $8127, 11352, 21672$ and 27090.
Since
\begin{tabular}{|c|c|c|}
\hline 1 & \begin{tabular}{l}
\begin{tabular}{l}
8127 \\
6450
\end{tabular}
\end{tabular} & \begin{tabular}{l}
\begin{tabular}{r}
11352 \\
8127
\end{tabular}
\end{tabular} \\
\hline 1 & 1677 & 3225 \\
\hline & 1548 & 1677 \\
\hline 12 & 129 & 1548 \\
\hline & & 1548 \\
\hline & 129 & 0 \\
\hline
\end{tabular}

Therefore, \((8127,11352)=129\), and by Lemma 10 we have
\[
\{8127,11352\}=\frac{8127 \times 11352}{129}=715176
\]

Since
\[
\begin{array}{|r|l|}
715176 & 21672 \\
715176 & \\
\hline 0 & 21672
\end{array}
\]

Therefore, \((715176,21672)=21672\), and by Lemma 10 we have
\[
\{715176,21672\}=\frac{715176 \times 21672}{21672}=715176
\]

Since
\[
2\left|\begin{array}{r|r|}715176 & 27090 \\ 704340 & 21672 \\ \hline 10836 & 5418 \\ 10836 & \\ \hline 0 & 5418\end{array}\right|
\]

Therefore, \((715176,27090)=5418\), and by Lemma 10 we have
\[
\{715176,27090\}=\frac{715176 \times 27090}{5418}=3575880
\]

Thus, we have
\[
\{8127,11352,21672,27090\}=3575880.
\]