Since $\operatorname{gcd}(d, 60)=5$, we know that $d=5^{i} d^{\prime}$ for some integer $i>0$ and some integer $d^{\prime}$ which is relatively prime to 60. Consequently, $d^{\prime}$ is a divisor of $(15!) / 5$; eliminating common factors with 60 gives that $d^{\prime}$ is a factor of $\left(7^{2}\right)(11)(13)$, which has $(2+1)(1+1)(1+1)=12$ factors. Finally, $i$ can be 1,2 , or 3, so there are a total of $3 \cdot 12=36$ possibilities.