Choosing n + 1 numbers among 1, …, 2n

Idea. Sort the numbers into boxes according to their odd part.

Boxes. Every $a\in\{1,\dots,2n\}$ can be written uniquely as $a=2^km$ with $k\ge0$ and $m$ odd. The odd part $m$ is one of the $n$ odd numbers $1,3,\dots,2n-1$.

Pigeonhole. The $n+1$ chosen numbers go into $n$ boxes, so two of them, $a=2^km$ and $b=2^lm$ with $a\ne b$, share the same odd part $m$. Then $k\ne l$, and if $k<l$ then $a$ divides $b$; otherwise $b$ divides $a$.

Sharpness. The $n$ numbers $n+1,n+2,\dots,2n$ contain no pair where one divides the other.

Lesson. Decompose objects into a canonical form and use one component as the pigeonhole box.