Let $f(n)$ be the sum of the reciprocals of all natural divisors of $n$. To find $f(pm)$, note that the sum of the reciprocal of all divisors not divisible by $p$ is $f(m)$, and the sum of the reciprocal of all divisors divisible by $p$ is $\tfrac{1}{p} f(m)$. Thus, $$f(pm) = f(m) + \dfrac{f(m)}{p}.$$We are given that $f(m)$ is an integer, so $f(pm)$ is an integer iff $f(m)$ is divisible by $p$. Observe that $f(m)$ is much smaller than $m$, so it is not divisible by $p$ because $p>m$. We can conclude that $f(pm)$ is not an integer, or $pm$ is not good, as desired.