The function $f$ from the set $\mathbb{N}$ of positive integers into itself is defined by the equality
$f(n)=\sum_{k=1}^{n} \gcd(k,n),\qquad n\in \mathbb{N}$
a) Prove that $f(mn)=f(m)f(n)$ for every two relatively prime ${m,n\in\mathbb{N}}$.
b) Prove that for each $a \in\mathbb{N}$ the equation $f(x)=ax$ has a solution.
c)[ Find all ${a\in\mathbb{N}}$ such that the equation $f(x)=ax$ has a unique solution.
ISL 2004-N2
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You spin my head right round right round,
When you go down, when you go down down......(-$from$ "$THE$ $UGLY$ $TRUTH$" )
When you go down, when you go down down......(-$from$ "$THE$ $UGLY$ $TRUTH$" )