proof that the sum of digits of natural number are divisible by 3 iff the number is

Im trying to prove that every natural number is divisble by three if and only if the sum of its digits are divisible by three.

First i proved by induction that $10^n-1$ is divisible by 9 (and therefore 3) so i could use this in the next step, would this be valid...

ANy natural number can be given a decimal representation

Let $s\in N$

Let $k_0,k_1...k_n\in${${0,...10}$}

$S=k_0 +10k_1+100k_2+...k_n10^n$

$= 9k_1 + 99k_2 + ... + k_n(10^n-1) + (k_0+k_1+...k_n)$

All terms are divisible by 3 but the sum of the digits in S. Therefore Dividing S by 3 will leave the same remainder as dividing the digits bys 3.

Is this a valid proof of the claim? Thank you for your time

Answer

It's fine, except that after the last $=$ sign you should have written $$9k_1+99k_2+\cdots+\overbrace{99\ldots9}^{n\text{ times}}k_n+(k_0+k_1+\cdots+k_n).$$