Rule: The last digit is even.
Example: 538,756. The last digit is a 6, and 6 is an even number. So, this number is divisible by 2.

Rule: The sum of the digits is divisible by 3.
Example: 9,213. The sum of the digits is 9+2+1+3=15. Since 15 is divisible by 3, so is this number.

Rule: The last 2 digits of the number form a number divisible by 4.
Example: 628. The last two digits are 28. The number 28 is divisible by 4.

Rule: The number ends in 0 or 5.
Example: 4,325,605. The number ends in a 5, so it's divisible by 5.

Rule: The number must be divisible by BOTH 2 and 3. See the rules for 2 & 3 above.
Example: 732

Rule: Take the last number and double it. Subtract that from the rest of the number. If you get an answer divisible by 7 (including zero), then the original number is divisible by seven. If you don't know the new number's divisibility, you can apply the rule again.

Example 203.
* The last digit is 3. Double it – 3x2=6.
* Subtract 6 from the rest of the number, which was 20. 20-6=14.
* Check to see if the difference (14 in this case) is divisible by 7.
* Since 14 is divisible by 7, so is 203.