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[a, b] + [a, c] + [b, c] = [a, b, c] + (a, b, c)

Source: 2021 Dürer Math Competition Regional E4 E+1 https://artofproblemsolving.com/community/c2773609_2021_

January 7, 2022
number theoryleast common multiplegreatest common divisorLCMGCD

Problem Statement

Determine all triples of positive integers a,b,ca, b, c that satisfy
a) [a,b]+[a,c]+[b,c]=[a,b,c][a, b] + [a, c] + [b, c] = [a, b, c].
b) [a,b]+[a,c]+[b,c]=[a,b,c]+(a,b,c)[a, b] + [a, c] + [b, c] = [a, b, c] + (a, b, c).
Remark: Here [x,y[x, y] denotes the least common multiple of positive integers xx and yy, and (x,y)(x, y) denotes their greatest common divisor.
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