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find min [lcm(a,b)+lcm(b,c)+lcm(c,a)]/[gcd(a,b)+gcd(b,c)+gcd(c,a)]

Source: Lotfi Zadeh Olympiad 2021, Problem 3

December 28, 2021

number theoryLCMGCDLotfi Zadeh MO

Problem Statement

Find the least possible value for the fraction

\frac{lcm(a,b)+lcm(b,c)+lcm(c,a)}{gcd(a,b)+gcd(b,c)+gcd(c,a)}

over all distinct positive integers

a, b, c

. By

lcm(x, y)

we mean the least common multiple of

x, y

and by

gcd(x, y)

we mean the greatest common divisor of

x, y