RECURRING DECIMALS I0I

The same principle can be applied to other cycles. In the
cycle for 1 we have 076923. The division of 100 by 13 leaves
a remalnder of 9 and this gives the relationship between
groups from which the cycle can be built up. Thus, instead
of multiplying in stages by 2 (as in the case of 142857) we
multiply by g:

07
63
567
5103
45927
413343
3720087
33480783

076923076 etc.

Similarly the principle can be extended if we divide 1000
instead of 100 and put the digits in groups of three. Dividing
1000 by 13 gives 76 and 12 remainder. The multiplier is
therefore 12 and proceedmg as before, multiplying succes-
sively by 12, we have

076
912

10944
131328

07652307 etc.

There is yet another way in which the cycle 142857 can be
built up, as follows:

 

7 7
7%50 359
7 X 502 17500
7 X 503 875000
7 X 504 43750000
7 X 503 2187500000

etc. 142857

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