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Prove that (ab+c, bc+a, ca+b)=(a,b,c)

Source: St. Petersburg MO 2001 Grade 9 Problem 4

March 1, 2023
St. Petersburg MOGCDnumber theoryDivisibilitygreatest common divisor

Problem Statement

Let a,b,c∈Z+a,b,c\in\mathbb{Z^{+}} such that (a2−1,b2−1,c2−1)=1(a^2-1, b^2-1, c^2-1)=1 Prove that (ab+c,bc+a,ca+b)=(a,b,c)(ab+c, bc+a, ca+b)=(a,b,c) (As usual, (x,y,z)(x,y,z) means the greatest common divisor of numbers x,y,zx,y,z) [I]Proposed by A. Golovanov
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