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Positive Cubes Divide Factorial Expression

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January 12, 2009
geometry3D geometryfactorialmodular arithmeticAMCAIMEnumber theory

Problem Statement

How many positive integer cubes divide 3!⋅5!⋅7! 3!\cdot 5!\cdot 7!? <spanclass=′latex−bold′>(A)</span> 2<spanclass=′latex−bold′>(B)</span> 3<spanclass=′latex−bold′>(C)</span> 4<spanclass=′latex−bold′>(D)</span> 5<spanclass=′latex−bold′>(E)</span> 6 <span class='latex-bold'>(A)</span>\ 2\qquad <span class='latex-bold'>(B)</span>\ 3\qquad <span class='latex-bold'>(C)</span>\ 4\qquad <span class='latex-bold'>(D)</span>\ 5\qquad <span class='latex-bold'>(E)</span>\ 6
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