Multiplying by a positive whole number
that’s less than
is easy. The result is simply
repeated. For example,
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and so on.

Most people know this, but what’s less well-known is that there’s also a neat trick to multiply larger numbers by Suppose
is a whole number with two digits. To work out
simply work out the sum of the digits of
and drop that sum in-between the digits. For example, let
The sum of its digits is
Dropping that sum between the digits gives
which is indeed equal to
There’s just one little caveat. If the sum of the digits of is
or larger, you need to carry a digit. In other words, you stick the right-most digit of the sum between the original digits and add the left-most digit of the sum to the original left-most digit. For example, if
then the sum of the digits is
We therefore stick a
between the original digits
and
and add a
to the
to get
which again is the correct result.
You can convince yourself that this trick always works using long multiplication. Suppose the digits of are
and
so
Long multiplication now tells us that
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from which our result easily follows.
Can you work out the trick for numbers with more than two digits? As a hint, here is the long multiplication when the digits of
are
up to
:
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