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Find x,y that gcd(x+y,mn)>1

Source: 8th European Mathematical Cup, Senior Category, Q1

December 26, 2019

number theory

Problem Statement

For positive integers

a

and

b

, let

M(a,b)

denote their greatest common divisor. Determine all pairs of positive integers

(m,n)

such that for any two positive integers

x

and

y

such that

x\mid m

and

y\mid n

M(x+y,mn)>1.

Proposed by Ivan Novak