How can you tell whether a number is divisible by another number
(leaving no remainder) without actually doing the division?
Why do "divisibility rules" work?

2   If the last digit is even, the number is divisible by 2

3   If the sum of the digits is divisible by three, the number is also

4   If the last two digits are divisible by 4, the number is also

5   If the last digit is a 5 or a 0, the number is divisible by 5.

6   If the number is divisible by both 3 and 2, it is also divisible
         by 6.

7   Take the last digit, double it, and subtract it from the rest of
         the number; if the answer is divisible by 7 (including 0),
         then the number is also

8   If the last 3 digits is divisible by 8, the number is also

9   If the sum of the digits is divisible by 9, the number is also.
          * This holds for any power of 3

10  If the number ends in 0, it is divisible by 10.

11   Subtract the sum of the even digits from the sum of the odd
        digits; if the difference, including 0, is divisible by 11,
        the number is also.

12   If the number is divisible by both 3 and 4, it is also
         divisible by 12.

13   Delete the last digit from the number, then subtract 9 times
        the deleted digit from the remaining number. If what is left
        is divisible by 13, then so is the original number.