= Solution
The <divergence> formula reduces the problem to the <ordinary differential equation> $rf'+nf=0$ on $r>0$. Multiplication by $r^{n-1}$ makes it an exact derivative:
$$
\frac{d}{dr}(r^nf(r))=r^{n-1}(rf'(r)+nf(r))=0.
$$
Since the interval $r>0$ is connected, $r^nf(r)$ is a single constant $C$. Conversely, substituting $f=Cr^{-n}$ into the <divergence> formula gives zero. Thus
$$
\boxed{\mathbf F(\mathbf x)=C\,\frac{\mathbf x}{|\mathbf x|^n}\quad(\mathbf x\ne0).}
$$
This proves both existence and uniqueness up to the constant within the specified radial class. For $n=1$, the two punctured half-lines still share this same constant because the stipulated coefficient is one <function> of $r=|x|$.
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