55. Give an argument to show that the 10-digit ISBN error-detection method will detect a digits. Does the same argument work for the fourth transposition error involving the first and third and sixth digits?
55.
For the transposition to go undetected, it must be the case that the difference of the correct number and the incorrect number is evenly divisible by 11. That is, is divisible by 11. This reduces to is divisible by 11. But is divisible by 11 only when is divisible by 11 and this happens only when . In this case, there is no error. The same argument works for the fourth and sixth digits.