Recently ran across a pointer to this "mind-reader" application. It's obvious how it works, which leads to some interesting recreational math in the area of how you can tell what divides into what.
The mind-reader thing is obvious; if you take any two-digit number, add the digits together, and subtract them from the original number, the result is divisible by 9. For example:
Almost like magic, isn't it? The whole area of divisibility leads to lots of mathemagical fun. Most people know that you can check whether a number is divisible by 3 just by adding up its digits. For example, consider 5543; 5+5+4+3 is 17, which isn't divisible by 3, so 5543 isn't either. But 5544 is. I seem to remember learning this in tenth grade or thereabouts.
If you think about it, the same is true of 9 (which is related to why the puzzle mentioned above works). And if you really scrunch up your forehead and think about it (this happened to me in a fit of insomnia), in a base-X numbering system, X - 1 and anything that divides it will have this property; this works for 9 and 3 in our base-10 system.
So for hexadecimal numbers, this is true of 15 ("F"), 5, and 3. And if anyone were demented enough to adopt a base-13 numbering system, it would be true of 2, 3, 4, 6, and 12.