SILK-ROAD 2002 Problem 4

Observe that the fraction $\frac{1}{7}=0,142857$ is a pure periodical decimal with period $6=7-1$ ,and in one period one has $142+857=999$ .For $n=1,2,\dots$ find a sufficient and necessary condition that the fraction $\frac{1}{2n+1}$ has the same properties as above and find two such fractions other than $\frac{1}{7}$ .