|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 non-adjacent digits. The difference between the correct entry and the entry with the transposed digits is equal to 10p-1 where p is the number of places separating the transposed digits. For adjacent digits p=1, so the difference equals 101-1 = 9. For digits separated by 2 places the difference is 102-1 = 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.
6. One interesting characteristic of 9 is the ease with which you can calculate it's multiples. The picture below illustrates the grade school favorite 'digital calculator': to multiply 4 by 9 hold out your hands with the fourth finger bent down. The number of digits to the left of the bent finger tells you the tens (3) and the number of digits to the right of the bent finger tells you the ones (6) for a result of 36.