Let $X$ be a projective surface. Let $D$ be a Cartier divisor which is locally defined by $f$ at a point $P$. Let $Z$ be a closed subscheme of $X$ only support at point $P$, and locally its ideal sheaf is of the form $(f,g)$, for some $g \in \mathcal{O}_{X,P}$. Let $\mu_P(Z)$ denote the local multiplicity of $Z$ at $P$.
Now my question is that can we find a Cartier divisor $E$ such that $E$ is locally defined by $g$ at $P$ (so that $Z$ is locally cut out by $D$ and $E$) and $D.E=\mu_P(Z)$? Intuitively, we just need the intersection $D.E$ concentrate at $P$ and no other points.
I feel like this should not be too hard (if it is true) or too restrictive (if it is conditionally true), since we have a lot of freedom at other points in support of E to avoid support of D. But I am not sure how to make this intuition rigorous.
Any comment/hint/reference is appreciated!
Take $X$ to be the projective plane and $D$ to be a smooth conic. Let $Z$ be a point on $D$ (so $\mu_P(Z)=1$). If $E$ is any divisor on $X$, one has $(E\cdot D)$ even, so you have an example where what you want fails.