TITLE: About the decimal period of $\frac 17$
QUESTION [7 upvotes]: It is easy to verify that $$\frac 17=\frac {142857}{999999}$$ where $142857$ is the decimal period of $\frac 17$. 
This period, which has six different digits, has the property that when multiplied by $1,2,3,4,5,6$, the respective products have the same six different digits in different position (Clearly, multiplied by $7$ must give $999999$).
Is $ 142857 $ the only six-digit number that has this property?

REPLY [6 votes]: Thus will be true for the period of $1/n$ iff the order of $10 \bmod n$ is $6$, but you'll have to consider different sets of multipliers.
For $n<100$, the examples are $n=7, 13, 21, 39, 63, 77, 91, 97$. (*)
For $n=13$, the number is $076923$ (let's accept this as having six digits) and there are two cycles: one for the multipliers $1,3,4,9,10,12$ and one for $2,5,6,7,8,11$.
(*) Apparently, there are only $53$ examples; see A059892 and A226477.