please forgive my english because I am from KUrdistan /Iraq.
In a more simple way, what you wrote means
(i) the sum of the digits of a number is divisible by 3 if and only if the number is divisible by 3.
(ii) the sum of the digits of a number is divisible by 9 if and only if the number is divisible by 9.
(i) Let a number divisible by 3 be then
The number in front of the mod is the rest after dividing by the modulo number.
We can write any number as .
Remark that .
Therefore and .
And you have your answer that is
(ii) Since you can draw the same conclusions.