Prove that if one chooses more than n numbers from the set then two of them are relatively prime.
Partition them as $\displaystyle \{1,2\},\ \{3,4\},\ \hdots,\{2n-1,2n\}$. Since there are $\displaystyle n$ parts, one part must contain two of the numbers, and hence two of the numbers are adjacent. Since adjacent numbers are relatively prime we are done.