jmc

$$3240=2^3\cdot 3^4\cdot 5^1$$A positive divisor of 3240 is a multiple of 3 when it has a prime factorization in the form $2^a\cdot 3^b\cdot 5^c$ where $0\le a\le 3$, $1\le b\le 4$, and $0\le c\le 1$. There are $4\cdot 4\cdot 2=\boxed{32}$ choices for $a$, $b$, and $c$, giving the number of positive divisors of 3240 that are multiples of 3.