To prevent 3 numbers from adding to a multiple of 3 you must have 3 numbers with a remainder that does not add to multiples of 3 – given that the remainders can either be 0, 1 or 2 the way to prevent this is to have numbers with remainders 0,0,1 0,1,1 0,0,2 0,2,2 1,1,2 or 1,2,2.
When picking five numbers you cannot do this as you will add a number that allows for the multiple of 3.
With 4 numbers, you can avoid this problem, as you can put 0,0,1,1 or 0,0,2,2.
However, with five numbers, you must add a 0, 1 or a 2 to the end of the remainders, making the results:
0,0,1,1,0
0,0,1,1,1
0,0,1,1,2
0,0,2,2,0
0,0,2,2,1
0,0,2,2,2
These all do not work as they all have 3 that add up to multiples of 3. The same is true with these for negatives as they just do the same things in the opposite direction.
Solution
163030
Problem
First name
Kai
School
Wilson's School
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