A Little Mathematical Proof

Did you know that if you take any non-negative integer, then subtract the sum of its digits, the answer will always be divisible by 9?

For example, today's date is 11/06. 1106-(1+1+6) = 1098, and 1098/9 = 122.

Curious why? Here's a more general proof [pdf], that the difference between a number and the sum of its digits (if written in base b) will always be divisible by (b-1).


  1. Nice piece of trivia. I'm wondering when this can ever be used in any practical sense or as a tool in a larger proof.

    See how long it takes you to figure out this problem:

    Two bicyclists starting 20 miles apart are heading towards each other at 10mph. At the same time, a fly that travels at 15mph starts from the front wheel of the southbound bicycle and flies to the front wheel of the northbound one, then turns around and flies to the front wheel of the southbound one again, and continues in this mane till he is crushed between the two front wheels. What total distance did the fly cover?

  2. I have (unfortunately or not) seen too many of these to be tricked.

    But I liked one mathematician's answer to this. He gave the correct answer (15 mi), then when asked how he solved it, said, "oh, I just summed the infinite sum in my head..."