Example 8 The general term formula of the sequence $\left\{f_{n}\right\}$ is
$$
\begin{array}{l}
f_{n}=\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^{n}\right]\left(n \in \mathbf{Z}_{+}\right) . \\
\text {Let } S_{n}=\mathrm{C}_{n}^{1} f_{1}+\mathrm{C}_{n}^{2} f_{2}+\cdots+\mathrm{C}_{n}^{n} f_{n} .
\end{array}
$$

Find all positive integers $n$ such that $S_{n}$ is divisible by 8.
Solve: This problem is based on a property of the F-sequence:
$$
\mathrm{C}_{n}^{1} f_{1}+\mathrm{C}_{n}^{2} f_{2}+\cdots+\mathrm{C}_{n}^{n} f_{n}=f_{2 n} \text {. }
$$

Therefore, $8\left|S_{n} \Leftrightarrow 8\right| f_{2 n}$.
According to the recurrence relation,
$$
f_{n}=8 f_{n-5}+5 f_{n-6} \text {, and } 81 f_{6}=8 \text {, }
$$

Thus, $8\left|f_{n} \Leftrightarrow 6\right| n$.
Hence, $8\left|f_{2 n} \Leftrightarrow 6\right| 2 n \Leftrightarrow 3 \mid n$.