Problem 1170

Lemma 4 Let $10, \beta>0, (b_{1}, 10)=1, b_{1}>1$

Suppose the order of 10 modulo $b_{1}$ is $h$, then
(1) When $\alpha=\beta=0$, $\frac{a}{b}$ can be expressed as a pure repeating decimal, and the length of the repeating part is exactly $h$, that is
$$\frac{a}{b}=0 . \dot{a}_{1} \cdots \dot{a}_{h}$$
(2) When $\mu=\max (\alpha, \beta) \geqslant 1$, $\frac{a}{b}$ can be expressed as a mixed repeating decimal, where the non-repeating digits are exactly $\mu$ in number, and the length of the repeating part is exactly $h$, that is
$$\frac{a}{b}=0 . a_{1} \cdots a_{\mu} \dot{a}_{\mu+1} \cdots \dot{a}_{\mu+h}$$