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 $x$ are also multiples of $y$?

None, unless $x$ is a multiple of $y$; so let's suppose that $\frac xy$ is an integer. In that case, $ky$ divides $x$ if and only if $y$ divides $\frac xy,$ so the number of positive factors of $x$ that are also multiples of $y$ is the same as the number of positive factors of $\frac yx,$ which I suppose you know how to compute from the prime factorization of $\frac yx.$

How many positive factors of $60$ are also multiples of $4$?

Since $\frac{60}4=15,$ that's the same as asking, how many positive integers are factors of $15$? The $4$ positive factors of $15,$ namely $1,3,5,15,$ correspond to the $4$ positive factors of $60$ which are multiples of $4,$ namely, $4,12,20,60.$
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.