$Problem:$ If a vector function $V=V(x,y,z)$ is not irrotational, show that if there exists a scalar function $g=g(x,y,z)$ such that $gV$ is irrotational, then $$V\cdot (\nabla \times V )=0$$
Remember, $V$ Irrotational $\iff \nabla\times V=0$
Since $\nabla\times V\not=0$, $\nabla\times V$ is a vector that is orthogonal to $V$, and so $V\cdot \nabla\times V=0$ by definition of the dot product.
I am totally unconvinced this can be the answer as nowhere did I use the fact that there exists a scalar function $g$ in my solution.
Any help is greatly appreciated!