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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?

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