CN-TST 2012 Problem 5

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 $m < n$ , we call $n$ a good number. Prove that for any positive integer $k$ , there are only finitely many good numbers not divisible by $k$ .