Problem 774

Solution:
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.