I realized the other day that in the decimal representation of $\frac{n}{7}$, where $n=\{1,2,3,4,5,6\}$, the decimal part always has the same sequence of six digits but shifted to start at a different number. Why is that?
To demonstrate:
$\frac{1}{7} = 0.142857...$
$\frac{2}{7} = 0.285714...$ (shifted right 2)
$\frac{3}{7} = 0.428571...$ (shifted right 1)
$\frac{4}{7} = 0.571428...$ (shifted right 4)
$\frac{5}{7} = 0.714285...$ (shifted right 5)
$\frac{6}{7} = 0.857142...$ (shifted right 3)