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1/x+1/y+1/[x, y]+1/(x, y)=1/2

Source: JBMO Shortlist 2007 N1

October 14, 2017
least common multiplegreatest common divisorJBMOnumber theory

Problem Statement

Find all the pairs positive integers (x,y)(x, y) such that 1x+1y+1[x,y]+1(x,y)=12\frac{1}{x}+\frac{1}{y}+\frac{1}{[x, y]}+\frac{1}{(x, y)}=\frac{1}{2} , where (x,y)(x, y) is the greatest common divisor of x,yx, y and [x,y][x, y] is the least common multiple of x,yx, y.
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