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Three LCM equations

Source: Problem #25 2016 AMC 10A

February 3, 2016
number theoryleast common multipleAMCAMC 10AMC 10 A2016 AMC 10A

Problem Statement

How many ordered triples (x,y,z)(x,y,z) of positive integers satisfy lcm(x,y)=72,lcm(x,z)=600\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600 and lcm(y,z)=900\text{lcm}(y,z)=900?
<spanclass=′latex−bold′>(A)</span> 15<spanclass=′latex−bold′>(B)</span> 16<spanclass=′latex−bold′>(C)</span> 24<spanclass=′latex−bold′>(D)</span> 27<spanclass=′latex−bold′>(E)</span> 64<span class='latex-bold'>(A)</span>\ 15\qquad<span class='latex-bold'>(B)</span>\ 16\qquad<span class='latex-bold'>(C)</span>\ 24\qquad<span class='latex-bold'>(D)</span>\ 27\qquad<span class='latex-bold'>(E)</span>\ 64
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