The Stacks project

Exercise 111.9.4. Let $A = k[x, y]_{(x, y)}$ be the local ring of the affine plane at the origin. Make any assumption you like about the field $k$. Suppose that $f = x^3 + x^2y^2 + y^{100}$ and $g = y^3 - x^{999}$. What is the length of $A/(f, g)$ as an $A$-module? (Possible way to proceed: think about the ideal that $f$ and $g$ generate in quotients of the form $A/{\mathfrak m}_ A^ n= k[x, y]/(x, y)^ n$ for varying $n$. Try to find $n$ such that $A/(f, g)+{\mathfrak m}_ A^ n \cong A/(f, g)+{\mathfrak m}_ A^{n + 1}$ and use NAK.)