CN-TST 2022 Problem 24

Given a positive integer $n$ , let $D$ be the set of all positive divisors of $n$ . The subsets $A,B$ of $D$ satisfies that for any $a \in A$ and $b \in B$ , it holds that $a \nmid b$ and $b \nmid a$ . Show that $$\sqrt{|A|}+\sqrt{|B|} \le \sqrt{|D|}.$$