Challenge 291 : A Choice Challenge
(a) If you select any five positive integers it will always be possible to find at least one group of three of them that sum to a multiple of three. For example, if I selected 3, 6, 8, 11, 16 then I could choose 3, 8 and 16 – with a sum of 27 – or I could choose 6, 11 and 16 – with a sum of 33.
There are other possible groups of three I could use in this example but they will all contain 16.
Why must this be?
(b) For a further (quite extreme) challenge, if I wanted to be able always to select four numbers from any group of n positive integers so that the sum of the four numbers would be a multiple of four, what is the smallest possible value of n?