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functional with f (mn) = f (m) f (n) / f (d)

Source: OIFMAT II 2012 day 1 p2 - Chilean Math Forum FMAT Olympiad https://artofproblemsolving.com/community/c2484778_oifmat

September 27, 2021
number theoryfunctionalfunctional equation in N

Problem Statement

Find all functions f:N→N f: N \rightarrow N such that: ∙\bullet f(m)=1  ⟺  m=1 f (m) = 1 \iff m = 1 ; ∙\bullet If d=gcd⁡(m,n) d = \gcd (m, n) , then f(mn)=f(m)f(n)f(d) f (mn) = \frac {f (m) f (n)} {f (d)} ; and ∙\bullet ∀m∈N \forall m \in N , we have f2012(m)=m f ^ {2012} (m) = m .
Clarification: fn(a)=f(fn−1(a))f^n (a) = f (f^{n-1} (a))
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