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Two mutually non-divisible subsets

Source: 2022 China TST, Test 4 P6

April 30, 2022

number theoryDivisibilityDivisors

Problem Statement

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|}.

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