Given two positive integer numbers m,n, show that m/n can't have period b−1 in base b.

Question

Given two positive integer numbers $m,n$, show that $\frac{m}{n}$ can't have period $b-1$ in base $b$.

MY ATTEMPT: wlog $m<n$, then suppose $\frac{m}{n}=0,a_1a_2...a_{b-1}$. Multiplying both sides by $b^{b-1}-1$ we obtain $\frac{m(b^{b-1}-1)}{n}=a_1...a_{b-1}$ with $a_i \in {0,1,2,...,b-1}$. How can I conclude?

Answer 1

I may miss something...

But $\overline{0,123456789123456789...}^{10}$ is a rational having a period equal to $9$ written in base $10$.