Since
\begin{array}{|c|c|c|} \hline 1 & \begin{array}{l} \begin{array}{l} 8127 \\ 6450 \end{array}
\end{tabular} & \begin{array}{l} \begin{array}{r} 11352 \\ 8127 \end{array}
\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
$$\{8127,11352,21672,27090\}=3575880.$$