Can You Figure Out This Math Magic Trick?

A former boss of mine used to love astounding our students with this seemingly simple math magic trick.  First, he'd have you pick a number from 1 to 63.  He would show you each card, asking you if your number appeared on that card.  After showing you all six cards, he was always able to pinpoint your number within seconds.

I like to think I'm pretty good at math puzzles and tricks, but I couldn't figure this one out until he explained it to me.  Can you figure out the trick?