Bob often finds it tedious to manually do long division to test for divisibility.
One day, he notices that if you take the sum of all the digits of a number, and this sum is divisible by 3 then the original number was also divisible by 3.
For example, take the number 27. 2+7 = 9, 9 is divisible by 3, thus Bob concludes that 27 is also divisible by 3.
(i) Does this trick work for all positive integers? If so, prove it.
(ii) Can you come up with a similar trick for numbers divisible by 11? Prove that it works for all positive integers.
