Basic divisibility of large numbers.

So I'm just going through KhanAcademy to refresh my basic pre-arithmetic and although it's embarassing I thought I'd get this thing checked up just for safety:

https://www.khanacademy.org/math/arithmetic/factors-multiples/divisibility_tests/v/divisibility-tests-for-2--3--4--5--6--9--10

In this video we check whether we can divide large numbers into 2, 3, 4, 5, 6, 9, 10. For checking whether the number was divisible by 2 it was simply whether the last number was even or odd. For 3 it was adding up all the digits in the number and seeing whether that was divisible by 3. My question is, does this method work for every number? Just add all the digits up and see whether it's divisible by the number? He used other methods for the preceeding numbers but I carried on with the adding of all the digits method and it seemed to work for the few I tried.

I appreciate any help, thanks a lot.

EDIT: Upon further tests it seems apparent that it's not. My bad.