What is so magical about the number nine? A number is divisible by 9 if and only if the sum of its digits are divisible by 9. Certain 'mental tricks' are done because a number minus the sum of its digits are divisible by 9, see http://www.milaadesign.com/wizardy.html for example.

Well it turns out that 9 is only magical in base 10.

Show that whenever a number is represented in base b, it is divisible by b-1 if and only if the sum of its digits are divisible by b-1.

For example, if we used base 9, then writing 143 in base 9 is (143)₉ = 1×9² + 4×9¹ + 3×9⁰ = 81 + 36 + 3 = 120 (in base 10). And this number is divisible by 8 because the sum of its digits (in base 9) are divisible by 8. (1 + 4 + 3 = 8.)

Also, a number written in base 9 will be divisible by 4 or 2 if and only if the sum of its digits are divisible by 4 or 2 respectively.
(33)₉ = 3×9¹ + 3×9⁰ = 27 + 3 = 30 is divisible by 2 because 3 + 3 = 6 is divisible by 2.

That is, factors of b – 1 share the property that a number represented in base b, it is divisible by a factor of b-1 if and only if the sum of its digits are divisible by that factor of b-1.

We know in base 10 that a number is divisible by three if and only if the sum of its digits are divisible by 3. This is only because 3 is a factor of 9, which is one less than our base 10.