Jim Phillips ([log in to unmask]) wrote:
: Can anyone explain why the works?  I know that any number multiplied by 9
: gives a number whose digits add up to 9, but why?  Is this an inevitable
: outcome of our base 10 number system?  Would a base 8 number system would
: the same for the number 7?

Assume a 2 digit number:

    (x * 10 + y) * 9 = w * 10 + z

    (x * (9 + 1) + y) * 9 = w * (9 + 1) + z
    x * 9 * 9 + x * 9 + y * 9 = w * 9 + w + z

Taking modulus by 9:

    0 == w + z, so w + z is a product of 9.

(This is how "casting out 9s" works.)

Your conjecture about 8 and 7 is true too.