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gcd of a^2+b^2-nab and a+b divides n+2 , when a,b are coprime positive

Source: Mexican Mathematical Olympiad 1988 OMM P5

July 27, 2018

number theorycoprimegreatest common divisor

Problem Statement

a

and

b

are coprime positive integers and

n

an integer, prove that the greatest common divisor of

a^2+b^2-nab

and

a+b

divides

n+2