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Prove that there exist 2011 consecutive amazing numbers

Source: Middle European Mathematical Olympiad 2011 - Team Compt. T-8

September 6, 2011
modular arithmeticgreatest common divisorarithmetic sequencenumber theoryrelatively primealgebrasystem of equations

Problem Statement

We call a positive integer nn amazing if there exist positive integers a,b,ca, b, c such that the equality n=(b,c)(a,bc)+(c,a)(b,ca)+(a,b)(c,ab)n = (b, c)(a, bc) + (c, a)(b, ca) + (a, b)(c, ab) holds. Prove that there exist 20112011 consecutive positive integers which are amazing.
Note. By (m,n)(m, n) we denote the greatest common divisor of positive integers mm and nn.
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