jmc

To express the number $0.42\overline{153}$ as a fraction, we call it $x$ and subtract it from $1000x$: \begin{array}{r r c r@{}l} &1000x &=& 421&.53153153153\ldots \\ - &x &=& 0&.42153153153\ldots \\ \hline &999x &=& 421&.11 \end{array}This shows that $0.42\overline{153}=\frac{421.11}{999}=\frac{42111}{99900}$.

So, $x=\boxed{42111}$.