Henry Ernest Dudeney/Modern Puzzles/98 - Curious Multiplication

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Modern Puzzles by Henry Ernest Dudeney: $98$

Curious Multiplication
I have frequently been asked to explain the following, which will no doubt interest many readers who have not seen it.
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?


Click here for solution

Sources

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