76 THE FASCINATION OF NUMBERS

(d) for 24 x 10, we have:

i 24 10
12 20
6 40
3 8o
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To obtain the answer, the fourth and fifth terms are taken.

In each case the terms taken in the right column coincide
with the odd numbers in the left column.

When multiplying by 24 (as in the last example) we are
really multiplying by (2%+22) and this reveals a relationship
between (a) the powers of 2 of which the multiplier is com-
posed, and (5) the numbers of the terms to be taken in the
other column. Thus if the multiplier is 2", we take the
(n+1)th term in the right column. If the multiplier is
(2" +-2%) then we take the (n+1)th and the (x+41)th terms
in the right column. In other words, the numbers of the
terms to be taken are always 1 greater than the indices repre-
senting the powers of 2 included in the multiplier.

Thus, to multiply by 72 this must first be reduced to its
components of powers of 2. This is done by taking the highest
power of 2 included (i.e. 26 =64) and expressing the remain-
der in the same way. Thus 72 =26+423, The indices of the
factors are 6 and 3, so that the terms to be taken in the right
column will be the seventh (6+1) and fourth (3+41), so:

72 6o (1st term)
36 120 end ,,
i 18 240 ged
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4 960 5th
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1920 Oth
3840  7th
The fourth and seventh terms together total 4320 and this
equals 72 x 60.
When we multiply by an odd number, say 17, we are in
fact multiplying by 2¢+41. Hence the first and fifth terms

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