AMC 12 2023b Problem 24

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

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

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

\textbf{(A)}~30 \textbf{(B)}~45 \textbf{(C)}~3 \textbf{(D)}~15 \textbf{(E)}~6