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Prove that if we randomly pick 39 numbers then there will always be 3 of them...

Prove that if we randomly pick 39 numbers then there will always be 3 of them with the property that their pairwise differences are divisible by 19.

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Answer #1

Let   be randomly picked numbers. Now we will look at the set modulo 19 (or, remainders modulo 19).So let

be the remainder set i,e .

Now there are 19 remainders and 39 numbers. Therefore by pigeon hole principle( box principle) there are at least 39/19 numbers same amongst   in the remainder set. 39/19>2, and it has to be an integer so we get 3 numbers having same remainder modulo 19 so their pairwise differences are divisible by 19. Proved.

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