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Prove that a number in the sequence $2,3,4,...,n \ (n>2$, is relatively prime to all other numbers if and only if it is a prime that exceeds $\displaystyle\frac{n}{2}$. Does such a prime always exist?

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If it's not a prime, all its divisors are there.
If it's not greater than n/2, than 2p is not greater than n, and it's in the list.

Every prime greater than n/2 trivially satisfies the condition. It always exists by Bertrand's postulate ($n/2 < p < 2\cdot n/2$).

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