Let K = 10A + B, where A is an element of {1,2,...,9} and B is an element of {0,1,...,9}. Then K is an elemnt of {10,11,...,99}. Suppose K - (A+B) is not a multiple of 9. Then K-(A+B) = 9n + p, where n and p are integers. Thus, 10A + B -A - B = 9n + p, so 9A = 9n + p, and hence 9(A-n)=p. If A=n, then p=0, and hence K is a multiple of 9. If A<>n, then p is a multiple of 9, and thus K is a multiple of nine.
QED, K is a multiple of 9.
(The great thing about the gixmo is that it moves the symbols every time you click it, so you think that you are getting a different answer.)