Definitely worth learning, however, there is in fact an alternative method that I think you can do with out modular arithmetic. Let's hope you at least know induction.

Lemma:

for all

k=1

This is clearly true.

(Induction Hypothesis) Suppose up to n,

Now we must show

But by induction hypothesis 9 divides the first term and clearly 9|9, so 9 divides the sum and the induction is complete.

Okay, so now pick up my old proof up at the point where I wrote out

Now instead of the modular argument, we will group nicely by coefficient.

So by our lemma

but this means 9 divides every term so 9 must divide the whole thing which is S(n)-n as desired. QED (again)