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PYPITIE LS AERLUEDENY ST RO EORIL 0 MARSERESALATT 0 O VEATV N

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98 THE FASCINATION OF NUMBERS

(a) Eventually in the process of multiplication a number
appears consisting entirely of a repetition of the
digit g.

(b) The digital root of each is g.

(a) A little consideration makes it at once apparent that if
the number derived from a cycle of recurring decimals is
multiplied by the number by which unity was divided to
obtain the original cycle, then a number consisting entirely
of nines must result. Thus, the number 142857, which is
derived from the division of unity by 7, becomes 999999 when
multiplied by 7.

Here it is necessary to realize that 1 does not equal
-142857. In fact it equals 142857 carried on interminably.
The cycle of digits in each case ends when the remainder
from a division is 1, for that is, so to speak, where we came in.
Since there must always be a remainder (unless we are divid-
ing by 2 or 5) the process is obviously endless. But since also
the cycle recommences on reaching the remainder 1, the
number represented by the cycle digits does not result exactly
from the division of 1 but results instead from the division of
‘9 (recurring to a limited extent, this extent varying for each
divisor).

Thus, I—;’ =-142857 (recurring indefinitely)

but —99%9—9 =+142857 (exactly)

Therefore, in order to obtain the number 142857 we
divide the number 999999. Consequently when 142857 is
multiplied by 7, we return to the original number 999999.

(b) So far as the digital root of cyclic decimals is concerned,
we may proceed as follows: From the foregoing it is seen that
if we take as the divisor of unity the number x and obtain as
the cyclic decimals the number » then (x) x () will give a
number consisting entirely of nines. Now the digital root of
any number consisting entirely of nines is itself g, and it