,
st

R ERRERRRATRRCoRTERo o

RECURRING DECIMALS 99

therefore follows that the digital root of (x)(y) must always
be 9.

One of the following propositions therefore applies to each
case:

(i) either x or y or both must have g as its digital root;
(1) both x and » must have g or 6 as their roots;

(i11) either x or y must have g as its root, and the other

must have 6 as its root.
From these propositions, the following may be derived:
(a) Unless the digital root of the divisor of unity is a mul-
tiple of g, then the digital root of the cyclic number
must be g.

(b) If the digital root of the divisor is g or 6, the root of the
cyclic number is 3, 6 or g.

(¢) If the digital root of the divisor is g, the root of the
cyclic number is 1, 3, 6 or g.

(d) The digital root of a cyclic number can never be 2, 4,
5, 77 or 8.

Another way of ‘building up’ the cycle -142857, and
other similar cycles, has, at first sight, the appearance of
magic. This is done by starting with the number 14 and then
doubling in successive stages, the results being moved two
places to the right at each stage.

1
e
'f

473

it

14
28
56
1IT2
224
448
896
1792
3584

142857142857142  etc.
Each of the above stages is shown as a multiplication, but
instead really results from a division, using 7 as the divisor