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gcd(a,b) x lcm(b,c), gcd(b,c)x lcm(c,a), ,gcd(c,a) x lcm(a,b)

Source: 2024 Czech and Slovak Olympiad III A p1

May 18, 2024
number theoryGCDLCM

Problem Statement

Let a,b,ca, b, c be positive integers such that one of the values gcd(a,b)⋅lcm(b,c),    gcd(b,c)⋅lcm(c,a),    gcd(c,a)−⋅lcm(a,b)gcd(a,b) \cdot lcm(b,c), \,\,\,\, gcd(b,c)\cdot lcm(c,a), \,\,\,\, gcd(c,a)-\cdot lcm(a,b) is equal to the product of the remaining two. Prove that one of the numbers a,b,ca, b, c is a multiple of another of them.
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