Henry Ernest Dudeney/Modern Puzzles/98 - Curious Multiplication/Solution

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Modern Puzzles by Henry Ernest Dudeney: $98$

Curious Multiplication
If a person can add correctly but is incapable of multiplying or dividing by a number higher than $2$,
it is possible to obtain the product of any two numbers in this curious way.
Multiply $97$ by $23$.
 97     23
 48    (46)
 24    (92)
 12   (184)
  6   (368)
  3    736
  1   1472
      ----
      2231
      ----
In the first column we divide by $2$, rejecting the remainders, until $1$ is reached.
In the second column we multiply $23$ by $2$ the same number of times.
If we now strike out those products that are opposite ton the even numbers in the first column
(we have enclosed these in brackets for convenience in printing)
and add up the remaining numbers we get $2231$, which is the correct answer.
Why is this?


Solution

In the first column, write down the successive remainders on division by $2$:

$1 \, 0 \, 0 \, 0 \, 0 \, 1 \, 1$

which, when reversed, becomes:

$1 \, 1 \, 0 \, 0 \, 0 \, 0 \, 1$

This is $97$ in binary notation, or $2^0 + 2^5 + 2^6$.

In the second column, we take the numbers next to where the remainders are $1$, and get:

$23 \times 1 + 23 \times 2^5 + 23 \times 2^6$

which evaluates to $2231$.

Thus it is seen that the whole operation is being done in binary notation.


Sources

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