The rule to determine if a number is divisible by nine without using any division. Simply add up all the digits of the number. Repeat this until you come out with a one digit number. If this number is nine, the original number is divisible by nine.
Example:
9 * 98765 (yeah, I'm lazy) = 888885
8 + 8 + 8 + 8 + 8 + 5 = 45
4 + 5 = 9
8 * 36547 = 292376
2 + 9 + 2 + 3 +7 +6 = 29
2 + 9 = 11
1 + 1 = 2
I learned this back in the day from watching Square One on PBS.
On the show it had a little ditty that went along with it:
Nine, nine, nine,
Fantastic number nine.
Perfectly efficient,
It works out every time.
Nine, nine . . .
I can't believe I still remember that.
This trick is similar to how to find out if ANY number is divisible by eleven.