This problem is from Zorich's Mathematical Analysis, Vol. I (Problem No. 8.6.6.2.(b), page 515).
Let $D$ be a domain in $\mathbb{R}^m$ and let $\phi, \psi ∈ C^{(k)}(D;\mathbb{R}), k\geq 1$. Suppose that whenever $\phi (x)=0$ we have $\psi(x)=0$ in the domain $D$. Show that if $grad\, \phi\neq 0$ in $D$, then there is a function $\theta ∈ C^{(k-1)}(D;\mathbb{R})$ such that on $D$ we can write $\psi =\theta \cdot \phi$.
Thank you.