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66 THE FASCINATION OF NUMBERS

so that all the following expressions:

50,000 —5
7,000 +7
600—6

50-+5 are multiples of 11.

Therefore 57650 —5-+7—6+5 must result in a
multiple of 11, so that 57650 when divided by
11 must give the same remainder as when
(5—%+6—5) is divided by 11. In the example
given, the number is therefore not divisible.

Divisor 12 The tests of divisibility are the same as for Divi-
sors g and 4 in conjunction.

Davisor 7

13 'These tests are less well known, but strangely
enough are exactly the same for both divisors.
The procedure is to split the number to be
divided into groups of three digits starting from
the right. These groups are then added and
subtracted alternately until a number of 3 or
fewer digits remains. If this final number is
divisible by 7 or 13, then so is the original num-
ber; if the final number is o, then the original
number is divisible by both 7 and 13.

Thus 510,300 becomes +510—300, reducing to
210. This is divisible by 7 but not by 13, and
the original number will be seen to have the
same properties. Similarly, the number 947,700
is found to be divisible by 13. Again, the num-
ber 151,632 becomes +151 —632=—481 and
is therefore divisible by 13. It will be seen that
the final reduced number may be either posi-
tive or negative, but that the result is the same.

These tests became known after the discovery that
the geometric series:

2

1080 108 rodiixoit ete.