in which bases we have $\frac 1x=0.\bar x$?

Question

In base ten we obtain the fact:$$1/3=0.\bar3=0.3333333333333\dots$$I want to know in which other bases thre is a number like that and what is that number.

Answer 1

You're actually asking for which base $a$ there is a number $b$ such that:$$\frac 1b=\sum_{n=1}^\infty ba^{-n}$$ but this is a geometric sequence so we need to find $$\frac 1b^2=\frac{1/a}{1-1/a}=\frac 1{a-1}$$ you want$$b=\sqrt{a-1}$$ I guess you want $b$ to be an integer so you need a base which is one above a square of a natural number (such: $2,5,10,17,\dots$) and the corresponding $b$ is $\sqrt{a-1}$