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Volume and Surface area

Source: 2015 AMC10B #25, AMC12B #23

February 26, 2015
geometryAMCAMC 10 BAMC 12

Problem Statement

A rectangular box measures a×b×ca \times b \times c, where a,a, b,b, and cc are integers and 1≤a≤b≤c1 \leq a \leq b \leq c. The volume and surface area of the box are numerically equal. How many ordered triples (a,b,c)(a,b,c) are possible?
<spanclass=′latex−bold′>(A)</span>4<spanclass=′latex−bold′>(B)</span>10<spanclass=′latex−bold′>(C)</span>12<spanclass=′latex−bold′>(D)</span>21<spanclass=′latex−bold′>(E)</span>26 <span class='latex-bold'>(A) </span>4\qquad<span class='latex-bold'>(B) </span>10\qquad<span class='latex-bold'>(C) </span>12\qquad<span class='latex-bold'>(D) </span>21\qquad<span class='latex-bold'>(E) </span>26
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