This set me to wondering. We use base-10 arithmetic, and 9 is 1 less than 10. I'm not a mathematician but something yanked at my intuition here, so I wondered if there was a similar formula for eleven.
Indeed there is! Take the first digit of the test number, subtract the second, add the third, subtract the fourth, etc., etc., and if the resulting sum/difference is a multiple of eleven (including zero and negative numbers), the original number is divisible by eleven as well.
I can't prove this (in fact, I figured it out by brute force), but can only say it's worked in every case I've tried. I suspect that there are formulas for all numbers (say, seven or thirteen), but I'm at a loss what they would be, and I'm not inclined to beat my head against that wall.
Anyone have anything to contribute here?
