Divisibility combinatorics

Question

Let $A$ be a set of $1008$ positive integers bounded above by $2014$. It is then said that there must be two integers in $A$ such that one divides the other but I can't immediately see how to prove this. It feels like a pigeonhole principle sort of question but I can't set it up.

Answer 1

Let $A=\{a_1,\dots,a_{1008}\}$. Write $a_i=2^{h_i}r_i$ where $h_i\ge 0$ and $r_i$ odd. Since $a_i\le 2014$ then $r_i\le 2013$. Note that there are $1007$ possibilities $r_i$'s and you have $1008$ numbers then there are $i<j$ such that $r_i=r_j$ so $a_i|a_j$ or $a_j|a_i$.