question from Discrete Mathematics and Its Applications:

Consider the following "mind-reading" trick. Come up with a 3 digit integer, call it x. Then shuffle anyway you like the digits of x, to come up with the integer y. Then find |x-y|, and determine the three digits, say a,b,c that represent the resulting positive number. Then any two of the digits can reveal the third digit, almost always, since one can show that a+b+c 0 (mod 9). For example, if two of the digits were, say b=4, c=9, then the third one is 'a' such that a+13=0 (mod 9). Because 'a' is a digit between 0 and 9, the answer was 5. One occasion that is problematic is whenever b+c 0 (mod 9) in which case 'a' could be either 0 or 9.

Find the formal proof that explains why a+b+c 0 (mod 9).