Let k \in \mathbb{Z}_{>0} be the smallest positive integer with the property that k \frac{\operatorname{gcd}(x, y) \operatorname{gcd}(y, z)}{\operatorname{lcm}\left(x, y^{2}, z\right)} is a positive integer for all values 1 \leq x \leq y \leq z \leq 121. If k^{\prime} is the number of divisors of k, find the number of divisors of k^{\prime}.