This simple problem is an example of a very interesting phenomenon: if you have a large enough "universe" to consider, even randomly picked parts exhibit structured properties.

Let \(n \geq 2\) be an integer. Then, consider the integers

\[ \{3, 4, \cdots, 2n-1, 2n\}\ .\]

Show that, if you pick \(n\) from those, you always have two numbers that will share no divisors whatsoever.

In other words, show that there's two of those \(n\) numbers that are coprime.

Give it some thought...

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