(a) Prove that given any 5 integers, there exist three of them whose sum is divisible by 3.
(b) Prove that given any 17 integers, there exist nine of them whose sum is divisible by 9.
You can just look at the worst case scenario. For example, for part (a), the worst case scenario is if you have the numbers 0, 0, 1 and 1. If you add any one more number, you can find three numbers such that their sum is divisible by 3.