Take a whole number, for example 1746, and reverse it, obtaining for example 6471. Three non-obvious things:
- The difference of the number and its reverse (or vice versa) is divisible by 9.
- If the original number has an odd number of digits, the difference of the number and its reverse is divisible by 11 (and by 9, hence by 99).
- If the original number has an even number of digits, the sum of the number and its reverse is divisible by 11.
Divisibility by 9
The number 1746 can be written as ; in general an n–digit number can be written , where the di are the digits. Then the number minus its reverse, or vice versa if the reverse is larger, is
For instance, . That last number is obviously divisible by 9, and in the general case the result is divisible by 9 if is. But for all , so .
Divisibility by 11
As before, the difference is divisible by 11 if is, where in this case n is odd; let ; then . But , therefore if is even (odd). and have the same parity, so , and .
By similar reasoning, when is even, the sum is divisible by 11 if is. and have opposite parity, so , and .