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finite good numbers is not divisible by $k$

Source: 2012 China TST - Quiz 1 - Day 2 - P5

March 15, 2012
number theory proposednumber theory

Problem Statement

For a positive integer nn, denote by τ(n)\tau (n) the number of its positive divisors. For a positive integer nn, if τ(m)<τ(n)\tau (m) < \tau (n) for all m<nm < n, we call nn a good number. Prove that for any positive integer kk, there are only finitely many good numbers not divisible by kk.
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