Yes. To sum it up, every integer is congruent to the sum of their digits modulo 9. With congruences, this ends up being pretty easy to prove.

Let be some integer with the representation of its digits being:

Note two things:

(1)

(2) can be written as a sum of the powers of 10 (as with any other integer):

Now we apply congruences:

Can you figure it out from here?