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Turkey NMO 2007 1st Round - P26 (Number Theory)

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October 5, 2012
least common multiplegreatest common divisor

Problem Statement

Let cc be the least common multiple of positive integers aa and bb, and dd be the greatest common divisor of aa and bb. How many pairs of positive integers (a,b)(a,b) are there such that 1a+1b+1c+1d=1? \dfrac {1}{a} + \dfrac {1}{b} + \dfrac {1}{c} + \dfrac {1}{d} = 1?
<spanclass=′latex−bold′>(A)</span> 6<spanclass=′latex−bold′>(B)</span> 5<spanclass=′latex−bold′>(C)</span> 4<spanclass=′latex−bold′>(D)</span> 3<spanclass=′latex−bold′>(E)</span> 2 <span class='latex-bold'>(A)</span>\ 6 \qquad<span class='latex-bold'>(B)</span>\ 5 \qquad<span class='latex-bold'>(C)</span>\ 4 \qquad<span class='latex-bold'>(D)</span>\ 3 \qquad<span class='latex-bold'>(E)</span>\ 2
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