AMC 12 2023 B Problem 24

Suppose that a, b, c, and d are positive integers satisfying all of the following relations.

$$a b c d =2^{6} \cdot 3^{9} \cdot 5^{7}$$
$$\operatorname{lcm}(a, b) =2^{3} \cdot 3^{2} \cdot 5^{3}$$
$$\operatorname{lcm}(a, c) =2^{3} \cdot 3^{3} \cdot 5^{3}$$
$$\operatorname{lcm}(a, d) =2^{3} \cdot 3^{3} \cdot 5^{3}$$
$$\operatorname{lcm}(b, c) =2^{1} \cdot 3^{3} \cdot 5^{2}$$
$$\operatorname{lcm}(b, d) =2^{2} \cdot 3^{3} \cdot 5^{2}$$
$$\operatorname{lcm}(c, d) =2^{2} \cdot 3^{3} \cdot 5^{2}$$

What is \operatorname{gcd}(a, b, c, d)?

Answer Choices
A. 30
B. 45
C. 3
D. 15
E. 6