The number of bases in which 
is a repeating decimal
(actually, repeating 
-ary)
of length 
is the same as the number of fractions 
, 
, ..., 
which have reduced denominator 
. For example, in bases 2, 3, ..., 6,
1/7 is given by
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
which have periods 3, 6, 3, 6, and 2, respectively, corresponding to the denominators 6, 3, 2, 3, and 6 of
 |
(6)
|
