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Misplaced much?

Source: 2023 AMC 12B p24

November 15, 2023
2023 AMC 12BAMCAMC 12number theoryleast common multiplegreatest common divisor2023 AMC

Problem Statement

Integers a,b,c,da, b, c, d satisfy the following: abcd=26⋅39⋅57abcd=2^6\cdot 3^9\cdot 5^7 lcm(a,b)=23⋅32⋅53\text{lcm}(a,b)=2^3\cdot 3^2\cdot 5^3 lcm(a,c)=23⋅33⋅53\text{lcm}(a,c)=2^3\cdot 3^3\cdot 5^3 lcm(a,d)=23⋅33⋅53\text{lcm}(a,d)=2^3\cdot 3^3\cdot 5^3 lcm(b,c)=21⋅33⋅52\text{lcm}(b,c)=2^1\cdot 3^3\cdot 5^2 lcm(b,d)=22⋅33⋅52\text{lcm}(b,d)=2^2\cdot 3^3\cdot 5^2 lcm(c,d)=22⋅33⋅52\text{lcm}(c,d)=2^2\cdot 3^3\cdot 5^2
Find gcd(a,b,c,d)\text{gcd}(a,b,c,d) <spanclass=′latex−bold′>(A)</span> 30<spanclass=′latex−bold′>(B)</span> 45<spanclass=′latex−bold′>(C)</span> 3<spanclass=′latex−bold′>(D)</span> 15<spanclass=′latex−bold′>(E)</span> 6<span class='latex-bold'>(A)</span>~30\qquad<span class='latex-bold'>(B)</span>~45\qquad<span class='latex-bold'>(C)</span>~3\qquad<span class='latex-bold'>(D)</span>~15\qquad<span class='latex-bold'>(E)</span>~6
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