What is known about representations of numbers of the form $\frac{1}{n}$, where $n$ is odd?

Question

I calculated some decimal representations of numbers of the form $\dfrac{1}{n}$, where $n$ is odd, and most of them had a period that began immediately after the decimal point. One example of a number where period does not begin immediately after the decimal point is number $\dfrac{1}{25}=0.04000000...$, but that should be because $25$ is a multiple of $5$ and we are working in base $10=2 \cdot 5$..

What is known about decimal representations of numbers of the form $\dfrac {1}{n}$, if $n$ is odd?

What are necessary and sufficient conditions for such a number to not have a pre-period, that is, that the period of that number starts immediately after the decimal point?

What are the results in other bases?

Where to learn more about this topic?

Answer 1

If $n$ is odd, then its decimal expansin will have a pre-period if and only if $n$ is a multiple of $5$. More generally, a decimal expansion of a number $\frac mn$, with $m,n\in\mathbb N$ and $m$ and $n$ relatively prime, will have a pre-period if and only if $n$ and $10$ are relatively prime. There is, of course, a similar result concerning expansions in any other base.

Answer 2

See here https://en.wikipedia.org/wiki/Fermat%27s_little_theorem. Due to little Fermat there's a number $k$ such that $n$ divides $10^k-1$ if $n$ and $10$ are relatively prime. Let $l$ be the smallest such number. Then $$\frac{1}{n}=\frac{\text{some numerator}}{\underbrace{9999..9999}_{\text{$l$ of them}} }.$$ In that case $$\frac{1}{n}=0.\overline{\underbrace{000\text{some numerator}}}_{\text{$l$ digits in total}}.$$

Example: Let $n=77$, then $l=6$ and $$\frac17=\frac{12987}{999999}=0.\overline{012987}.$$