Here is a decimal expansion of $\frac{1}{34}$:
$$(1/34)_{10}=0.02941176470588235\overline{2941176470588235}\ldots$$
And here is a graphical representation of the 16-digit
"repetend," as a directed *repetend digit graph* (my terminology): 
$$(2,9,4,1,1,7,6,4,7,0,5,8,8,2,3,5)\;.$$
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![Digitsn34b10][1]
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I was exploring the digit-expansion of $1/n$ 
in base $b$&mdash;fixing $n$ while letting $b$ vary&mdash;and find it puzzling.
Here is an example, for $n=51$, and bases $b=5,\ldots,50$.
The top row shows base $b$, and underneath, the length of the repetend 
for $\frac{1}{51}$ in that
base:
$$
\left(
\begin{array}{cccccccccccccccc}
 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 &
   13 & 14 & 15 & 16 & 17 & 18 & 19 &
   20 \\
 16 & 16 & 16 & 8 & 8 & 16 & 16 & 16
   & 4 & 16 & 8 & 2 & 2 & 1 & 8 & 16
   \\
\end{array}
\right)
$$
$$
\left(
\begin{array}{ccccccccccccccc}
 21 & 22 & 23 & 24 & 25 & 26 & 27 &
   28 & 29 & 30 & 31 & 32 & 33 & 34 &
   35 \\
 4 & 16 & 16 & 16 & 8 & 8 & 16 & 16 &
   16 & 4 & 16 & 8 & 2 & 1 & 2 \\
\end{array}
\right)
$$
$$
\left(
\begin{array}{ccccccccccccccc}
 36 & 37 & 38 & 39 & 40 & 41 & 42 &
   43 & 44 & 45 & 46 & 47 & 48 & 49 &
   50 \\
 8 & 16 & 4 & 16 & 16 & 16 & 8 & 8 &
   16 & 16 & 16 & 4 & 16 & 8 & 2 \\
\end{array}
\right)
$$
It is evident that the repetend length is a factor of $17{-}1$; and $n=3 {\cdot} 17$.
I tried to understand when the repetend digit graphs were isomorphic,
but a pattern is not evident. For example, for $\frac{1}{51}$, for bases
$$b = 15,19,25,26,32,36,42,43,49 \;,$$
that graph is an octagon. Here are three of them:
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![Digitsn51][2]
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So here is a specific question:

> **Q**. Is it possible to predict which of the base-$b$ digit-expansions of $1/n$
result in isomorphic repetend digit graphs?
In particular, graphs which are cycles?
Perhaps specifically when $n$ is a prime?


  [1]: https://i.sstatic.net/M42Ur.jpg
  [2]: https://i.sstatic.net/THgjy.jpg