Some people having been talking about divisibility rules and why they work. For example, Sasha gave a proof for the "divisibility by 9" rule, which you should check out because it is pretty cool.

But anyways so I got to wondering just how many numbers have rules such as these? And why can we do this with some numbers, but not others?

I think many of us are aware of the rules for 2, 4, 5, 6, & 9. But what I just learned today is that this list kind of goes on forever. Here are some examples I found on Wikipedia, to help clarify what I mean:

**33** Rule: Add 10 times the last digit to the rest. EX: 627: 62 + 7 x 10 = 132, 13 + 2 x 10 = 33.

**79** Rule: Add 8 times the last digit to the rest. EX: 711: 1x8=8; 8+71=79

**989** Rule: Divide the number of thousands by 989. Multiply the remainder by 11 and add to last 3 digits. . EX: 21758: 21/989 Remainder = 21, 21 x 11 = 231; 758 + 231=989

(http://en.wikipedia.org/wiki/Divisibility_rule#Beyond_20)

They have 27 examples like this total, ranging mostly from 21 to 91, and then ending with 989. You have to wonder how one comes up with these crazy observations, and furthermore, how they are proved. Underneath this chart there is a section that mentions different ways of proving these, and I noticed one way is using modular arithmetic. I found this to be particular interesting since Andy mentioned in class today that we would be starting a modular arithmetic unit tomorrow.

Sooo… I was wondering, if any of you have seen this before? If you have, do you know if modular airthmetic serves as a proof for all divisibility rules, or just some of them? Is there some sort of algorithm that is used to come up with this crazy rules?

Thanks :)