A question asked to show that $\vec{F}=\frac{-yi+xj}{x^2+y^2}$ is not conservative. My intuition was that $\vec{F}$ could be rewritten as $\frac{1}{x^2+y^2}(-yi+xj)$ and $-yi+xj$ is not a gradient nor conservative.
I know that you can choose a curve (unit circle) and show that is in fact not conservative, my question is whether my intuition is correct, ie. if a vector field can be rewritten as a scalar function (not identically 0) times a non conservative field would it always result in a non-conservative field.