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HMMT Combinatorics 2019/8: a, b, gcd(a, b) different colors if all different

Source:

February 17, 2019

HMMTcombinatorics

Problem Statement

For a positive integer

N

, we color the positive divisors of

N

(including 1 and

N

) with four colors. A coloring is called multichromatic if whenever

a

b

and

\gcd(a, b)

are pairwise distinct divisors of

N

, then they have pairwise distinct colors. What is the maximum possible number of multichromatic colorings a positive integer can have if it is not the power of any prime?

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