Proof of the divisibility test for 9 and 3

It is a fairly well known test that, to check if an integer is divisible by 9 (or 3), you simply add the digits of the integer and see if the result is divisible by 9 (or 3). In this video, we see a proof of why this is the case!

[shairyl]

Fantastic tutorial. Helped me a lot. I just have one issue though. When you suppose the sum of the digits is a multiple of 9, isnt the defeating the purpose of proving it? If it is a multiple of 9 then ofcourse it is divisible by 3 and 9