MathDB
Multiplicative function

Source: IMO Shortlist 2004, number theory problem 2

March 23, 2005
functionnumber theorygreatest common divisorequationIMO Shortlist

Problem Statement

The function ff from the set N\mathbb{N} of positive integers into itself is defined by the equality f(n)=∑k=1ngcd⁡(k,n),n∈N.f(n)=\sum_{k=1}^{n} \gcd(k,n),\qquad n\in \mathbb{N}. a) Prove that f(mn)=f(m)f(n)f(mn)=f(m)f(n) for every two relatively prime m,n∈N{m,n\in\mathbb{N}}.
b) Prove that for each a∈Na\in\mathbb{N} the equation f(x)=axf(x)=ax has a solution.
c) Find all a∈N{a\in\mathbb{N}} such that the equation f(x)=axf(x)=ax has a unique solution.
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