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)