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.

A crowd of people, photo by Rob Curran on Unsplash

Problem statement

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...

If you need any clarification whatsoever, feel free to ask in the comment section below.

Solution

You can read the solution here to compare with your own solution. You can also use that link to post your own solution in the comments! Please do not post spoilers in the comments here.


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