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Multiplicative function for coprime m,n

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November 9, 2010
functionnumber theory proposednumber theory

Problem Statement

The function ψ:N→N\psi : \mathbb{N}\rightarrow\mathbb{N} is defined by ψ(n)=∑k=1ngcd⁡(k,n)\psi (n)=\sum_{k=1}^n\gcd (k,n).
(a)(a) Prove that ψ(mn)=ψ(m)ψ(n)\psi (mn)=\psi (m)\psi (n) for every two coprime m,n∈Nm,n \in \mathbb{N}. (b)(b) Prove that for each a∈Na\in\mathbb{N} the equation ψ(x)=ax\psi (x)=ax has a solution.
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