Example 5 Please convert $\frac{15}{308}$ into a decimal.
Since $308=4 \times 77=2^{2} \cdot 77, \varphi(77)=(11-1) \times (7-1)=60$.
We also have
$$\begin{array}{ll}
10 \equiv 10(\bmod 77), & 10^{2} \equiv 23(\bmod 77) \\
10^{3} \equiv 76(\bmod 77), & 10^{4} \equiv 67(\bmod 77) \\
10^{5} \equiv 54(\bmod 77), & 10^{6} \equiv 1(\bmod 77)
\end{array}$$

Therefore, in Lemma 5, we can take $a=15, b=308, \alpha=2, \beta=0$, $h=6, b_{1}=77$. By Lemma 5, we have $\frac{15}{308}=0 . a_{1} a_{2} a_{3} a_{4} \cdots a_{8}$, and after calculation, we have
$$\frac{15}{308}=0.04 \dot{8} 7012 \dot{9}$$