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find all triples (a,b,c)

Source: India tst 2003, p2

March 14, 2012
number theorygreatest common divisornumber theory proposed

Problem Statement

Find all triples (a,b,c)(a,b,c) of positive integers such that (i) a≤b≤ca \leq b \leq c; (ii) gcd(a,b,c)=1\text{gcd}(a,b,c)=1; and (iii) a3+b3+c3a^3+b^3+c^3 is divisible by each of the numbers a2b,b2c,c2aa^2b, b^2c, c^2a.
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