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Functional Equation with gcd and lcm

Source: Korea National 2013 #5

November 10, 2013
number theorygreatest common divisorleast common multiplefunctionnumber theory proposed

Problem Statement

Find all functions f:N→Nf : \mathbb{N} \rightarrow \mathbb{N} satisfying f(mn)=lcm⁡(m,n)⋅gcd⁡(f(m),f(n)) f(mn) = \operatorname{lcm} (m,n) \cdot \gcd( f(m), f(n) ) for all positive integer m,nm,n.
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