Finitely many 'good' numbers

For discussing Olympiad Level Number Theory problems
For a positive integer $n$, denote by $\tau (n)$ the number of its positive divisors. For a positive integer $n$, if $\tau(m) < \tau(n)$ for all positive integers $m<n$, we call $n$ a good number. Prove that for any positive integer $k$, there are only finitely many good numbers not divisible by $k$.