Prove that if one chooses more than n numbers from the set http://stuff.daniel15.com/cgi-bin/ma...202n%5Crbrace, then two of them are relatively prime.

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- Oct 16th 2009, 07:15 AMusagi_killerPrimes
Prove that if one chooses more than n numbers from the set http://stuff.daniel15.com/cgi-bin/ma...202n%5Crbrace, then two of them are relatively prime.

- Oct 16th 2009, 08:13 AMBruno J.
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.