Example 5 Given
$$
A=\left\{z \mid z^{18}=1\right\} \text { and } B=\left\{\omega \mid \omega^{48}=1\right\}
$$

are sets of complex roots of unity,
$$
C=\{z w \mid z \in A, w \in B\}
$$

is also a set of complex roots of unity. How many distinct elements are there in the set $C$? ${ }^{[3]}$
Notice that, $z=\cos \frac{2 k \pi}{18}+\mathrm{i} \sin \frac{2 k \pi}{18}(k \in \mathbf{Z})$ (18 distinct elements),
$\omega=\cos \frac{2 t \pi}{48}+\mathrm{i} \sin \frac{2 t \pi}{48}(t \in \mathbf{Z})$ (48 distinct elements),
$$
\begin{array}{l}
z \omega=\cos \frac{2 \pi(8 k+3 t)}{144}+\mathrm{i} \sin \frac{2 \pi(8 k+3 t)}{144} . \\
\text { Let } P=\{m \mid m=8 k+3 t, t \in \mathbf{Z}\} .
\end{array}
$$

By Bézout's theorem, $P=\mathbf{Z}$.
Therefore, the set $C$ contains 144 distinct elements.