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Part of 2013 Switzerland - Final Round

Problems(1)

{gcd(a,b),gcd(b,c),gcd(c,a), lcm(a,b), lcm(b,c), lcm(c, a)}= {2, 3, 5, 30, 60}

Source: Switzerland - 2013 Swiss MO Final Round p1

12/30/2022
Find all triples (a,b,c)(a, b, c) of natural numbers such that the sets {gcd(a,b),gcd(b,c),gcd(c,a),lcm(a,b),lcm(b,c),lcm(c,a)}\{ gcd (a, b), gcd(b, c), gcd(c, a), lcm (a, b), lcm (b, c), lcm (c, a)\} and {2,3,5,30,60}\{2, 3, 5, 30, 60\} are the same.
Remark: For example, the sets {1,2013}\{1, 2013\} and {1,1,2013}\{1, 1, 2013\} are equal.
number theorygreatest common divisorleast common multipleLCMGCD