Ways to test for divisibility by single-digit numbers in base 10:

1
This is here for completeness. Every integer is divisible by 1.
2
The last digit is 0, 2, 4, 6, or 8.
3
The sum of the digits is divisible by 3. (If you are dealing with a freaking huge number you can apply this recursively.)
4
The last two digits form a number which is itself divisible by 4.
5
The last digit is 0 or 5.
6
The sum of the digits is divisible by 3 and the last digit is 0, 2, 4, 6, or 8.
7
I defer to how to determine whether a number is divisible by 7.
8
Either:
  • The hundreds place is even, and the last two digits form a number which is divisible by 8.
  • orThe hundreds place is odd, and the last two digits minus 4 form a number which is divisible by 8.
9
The sum of the digits is divisible by 9.

Divisibility is not symmetric. a | b usually doesn't imply b | a. The property that a | a means that divisibility is reflexive.