DIVISIBILITY 67

where each term is multiplied by 102 to obtain
the next term, is remarkable for the fact that
either the addition to or subtraction from each
term of the number 1 will give a multiple of

7 Oor13.
103+41=1001 =7 XI3XI1I
10°—1=999999 =7 X 13 X 10989

10° 41 =1000000001 =7 X 13 X 10989011

The tests are, of course, useful only in respect of
large numbers. A certain type of number may,
however, be immediately identified as a mul-
tiple of both 7 and 13. Such a number consists
of the form ‘abcabc’ where obviously abc —abe =o.
Thus 176,176 is divisible by both 7 and 13. It
should be noted that in the form ‘abcabc’, either
a, b or ¢ or any two of them may be noughts.
17,017 can be read also as 017,017, and 6006
can be read as 006,006, so that each conforms
to the ‘abcabe’ form and can readily be identi-
fied as multiples of both 7 and 13.

Ordinary division of an apparently complex nature can be
very greatly simplified. Many processes in mathematics are
made easier by the expedients of borrowing and paying back.
In a somewhat different sense, if we wish to divide by a cer-
tain number, it may prove simpler to divide by another num-
ber and make any necessary adjustments later on or as we
proceed. In Chapter g it was commented upon that the
number g has certain peculiarities in multiplication by virtue
of its relationship to the number 10. We now have another
example in respect of division, for the fact emerges that in
order to divide any number by 9, we may instead divide by
10 and make elementary compensating adjustments as we
proceed. In fact, these adjustments are so very simple that
the whole process can be carried out mentally, by a method
first demonstrated by A. H. Russell.

 

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