Question 24
Suppose that $a,$ $b,$ $c,$ and $d$ are positive integers satisfying all of the following relations.
$$
\begin{aligned} abcd &= 2^6\cdot 3^9\cdot 5^7\\ \operatorname{lcm}(a,b) &= 2^3\cdot 3^2\cdot 5^3\\ \operatorname{lcm}(a,c) &= 2^3\cdot 3^3\cdot 5^3\\ \operatorname{lcm}(a,d) &= 2^3\cdot 3^3\cdot 5^3\\ \operatorname{lcm}(b,c) &= 2^1\cdot 3^3\cdot 5^2\\ \operatorname{lcm}(b,d) &= 2^2\cdot 3^3\cdot 5^2\\ \operatorname{lcm}(c,d) &= 2^2\cdot 3^3\cdot 5^2 \end{aligned}
$$
What is $\gcd(a,b,c,d)?$
$$
\begin{aligned} abcd &= 2^6\cdot 3^9\cdot 5^7\\ \operatorname{lcm}(a,b) &= 2^3\cdot 3^2\cdot 5^3\\ \operatorname{lcm}(a,c) &= 2^3\cdot 3^3\cdot 5^3\\ \operatorname{lcm}(a,d) &= 2^3\cdot 3^3\cdot 5^3\\ \operatorname{lcm}(b,c) &= 2^1\cdot 3^3\cdot 5^2\\ \operatorname{lcm}(b,d) &= 2^2\cdot 3^3\cdot 5^2\\ \operatorname{lcm}(c,d) &= 2^2\cdot 3^3\cdot 5^2 \end{aligned}
$$
What is $\gcd(a,b,c,d)?$