Question 103
Let $S$ be the set of all positive factors of $6000$. What is the probability that a random quadruple $(a, b, c, d) \in S^{4}$ satisfies
$$
\operatorname{lcm}(\operatorname{gcd}(a, b), \operatorname{gcd}(c, d)) = \operatorname{gcd}(\operatorname{lcm}(a, b), \operatorname{lcm}(c, d))?
$$
$$
\operatorname{lcm}(\operatorname{gcd}(a, b), \operatorname{gcd}(c, d)) = \operatorname{gcd}(\operatorname{lcm}(a, b), \operatorname{lcm}(c, d))?
$$