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I am asked:

Prove that if n+1 distinct numbers are selected from the first 2n positive integers {1,2,3,...,2n-1,2n} then at least two of the n+1 numbers are co-prime where n is a positive integer

I know 32 divides evenly into 64 and 96, but 32 is bigger than 16 and 20 so could not even divide. However 3+2=5 which can't get smaller.

How can I prove this?

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It might be easier to show that two of the numbers are adjacent. –  Dave Radcliffe Nov 13 '12 at 3:23
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1 Answer

up vote 2 down vote accepted

Divide the set into n disjoint subsets as such:

{1,2},{3,4},{5,6},...,{2n-1,2n}

By the pegionhole principle, if you select n+1 numbers 2 of them must be in one of the above subsets. These 2 are coprime. (This is beacuse for all positive integers i,i+1 are coprime)

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