1. Transposition of digits (checkbook problem)
The error caused by a transposition of digits is divisible by 9.
Assume we have two integers differing by a single transposition of adjacent digits. We can express them as strings of digits:
This works similarly for transpositions of nonadjacent digits. The difference between the ‘correct’ entry and the entry with the transposed digits is equal to 10^{p}1 where p is the number of places separating the transposed digits. For adjacent digits p=1, so the difference equals 10^{1}1 = 9. For digits separated by 2 places the difference is 10^{2}1 = 99. The interesting fact is that no matter which two digits are transposed the ‘error’ will always be divisible by 9. So, if you are reconciling your checkbook and you find a discrepancy that is divisible by nine this is the type of error you should suspect.
