When I first tried, I wondered how this could happen; but later I noticed I could simplify the 5 different numbers by what their remainder was from 3 (as 5 or 2 and 32 are basically the same for adding together) So, the reason it is impossible to not be able to make a multiple of 3 with 5 numbers is because:
If there are 3 of any type ( 3 numbers that have 0 remainder divided by 3, 3 numbers that have 0 remainder divided by 2, 3 numbers that have 3 remainder divided by 2, ) then the total will be divisible by 3.
If there is one of each type, then they will add up to 3.
5 numbers would have to be chosen because if it was 4, then these numbers could have been chosen :
1, 1, 2, 2
By chosing 3 of them, you cannot make them add up to a multiple of 3.
Solution
163067
Problem
First name
Wesley
School
Wilson's School
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