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AMC (AIMO validation set) / https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_24 / Suppose that a, b, c and d are positive integers satisfying all of the following relations. abcd=2⁶⋅3⁹⋅5⁷ lcm(a,b)=2³⋅3²⋅5³ lcm(a,c)=2³⋅3³⋅5³ lcm(a,d)=2³⋅3³⋅5³ lcm(b,c)=2¹⋅3³⋅5² …
Problem
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problem
Suppose that , , and are positive integers satisfying all of the following relations.
What is ?
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Suppose that a, b, c and d are positive integers satisfying all of the following relations. abcd=2⁶⋅3⁹⋅5⁷ lcm(a,b)=2³⋅3²⋅5³ lcm(a,c)=2³⋅3³⋅5³ lcm(a,d)=2³⋅3³⋅5³ lcm(b,c)=2¹⋅3³⋅5² lcm(b,d)=2²⋅3³⋅5² lcm(c,d)=2²⋅3³⋅5² What is gcd(a,b,c,d)?
Original LaTeX notation
Suppose that $a$, $b$, $c$ and $d$ are positive integers satisfying all of the following relations.
\[abcd=2^6\cdot 3^9\cdot 5^7\]
\[\text{lcm}(a,b)=2^3\cdot 3^2\cdot 5^3\]
\[\text{lcm}(a,c)=2^3\cdot 3^3\cdot 5^3\]
\[\text{lcm}(a,d)=2^3\cdot 3^3\cdot 5^3\]
\[\text{lcm}(b,c)=2^1\cdot 3^3\cdot 5^2\]
\[\text{lcm}(b,d)=2^2\cdot 3^3\cdot 5^2\]
\[\text{lcm}(c,d)=2^2\cdot 3^3\cdot 5^2\]
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