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Divisor multiplicity

Source: 2017 SDMO High School Problem 4

November 8, 2017
number theoryrelatively prime

Problem Statement

For each positive integer nn, let τ(n)\tau\left(n\right) be the number of positive divisors of nn. It is well-known that if aa and bb are relatively prime positive integers then τ(ab)=τ(a)τ(b)\tau\left(ab\right)=\tau\left(a\right)\tau\left(b\right). Does the converse hold? That is, if aa and bb are positive integers such that τ(ab)=τ(a)τ(b)\tau\left(ab\right)=\tau\left(a\right)\tau\left(b\right), then is it necessarily true that aa and bb are relatively prime? Either give a proof, or find a counter-example.
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