Question: I recently came upon the question that follows. 
“How many positive factors of 60 are also multiples of 4?”
Since 60 is relatively small I used a brute force method.
But I was wondering if there was a way you could do this using the prime factorization of 60.

Thanks!

Answer: 
How many positive factors of $n$ are also multiples of $m$?

Unless $m$ is a factor of $n$, there aren't any.

But suppose $m\mid n$. Then a multiple of $m$ is $md$, and $md\mid n$
iff $n/(md)\in\Bbb N$ iff $d\mid(n/m)$. So, the answer to the
question is "the number of positive factors of $n/m$".
Sure there is. It's more instructive to look at a bigger example. How many factors of $720$ are multiples of $4?$ Since $720=2^4\cdot3^2\cdot5,$ any factor of $720$ is of the form $2^a3^b5^c,$ where $0\le a\le4,\ 0\le b\le2,\ 0\le c\le1.$ However, for it to be a multiple 0f $4,$ we need $2,\le a$, so we have $3$ choices for $a$, $3$ for $b,$ and $2$ for $c,$ or $3\cdot3\cdot2=18$ in all.