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100 THE FASCINATION OF NUMBERS

of unity. Here it is necessary to remember that instead of
dividing into 1, we can divide into 10 or 100 and still get
the same digits, merely moving the decimal point afterwards.
We cannot divide 1 by 7 and so we normally bring down a
nought so that, in fact, we are dividing 10 by 7. If, however,
we extend the process and bring down two noughts together
we are really dividing 100 by 7. The important point is that
the resulting digits will be the same. A second point to consider
is that every number will divide into a larger number an
integral number of times and (unless it is also a factor of the
larger number) leave a remainder. A third point is that, in
dividing a large number involving many successive divisions,
the remainder from each division is important as providing
the basis for the next division.
We can now proceed as follows:

(a) We cannot divide 1 by 7, so we bring down two
noughts.

(b) Divide 100 by 7. This equals 14, and remainder 2.

(¢) Carry forward the remainder (2) and add two noughts.
Divide 200 by 7. This equals 28, and remainder 4.

This process can be best seen as follows:

Operation Result Remainder
1007 14 2

200 -7 28 4

400 -7 56 8

It is at once apparent why, having obtained the result of
the first division (giving 14) all that is necessary to obtain
successive answers is to double the previous answer, because
each successive remainder is double the previous remainder.

Algebraically this can be expressed: if %9 gives y and a

: O L k
remainder of 2, then — will give 2y and a remainder of 4.
x

Since the numerator is doubled, then both the result and the
remainder are also doubled.