A Monte Carlo approach is used to study the electron mobility in the Si/${\mathrm{Si}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Ge}}_{\mathit{x}}$ system at low temperatures. The diffusion constant is evaluated in near thermal equilibrium simulations and is converted to the mobility by use of the Einstein relation for a degenerate two-dimensional electron gas. A modulation-doped structure is considered, where the doped ${\mathrm{Si}}_{0.7}$${\mathrm{Ge}}_{0.3}$ layer provides channel electrons and is separated from the channel by an undoped ${\mathrm{Si}}_{0.7}$${\mathrm{Ge}}_{0.3}$ spacer layer. The electron density is evaluated as a function of spacer width and doping concentration. Electrons are assumed to be only in the lowest subband. Acoustic-phonon scattering and remote impurity scattering determine the possible mobility that can be reached. We find mobility values of 2.5\ifmmode\times\else\texttimes\fi{}${10}^{5}$ ${\mathrm{cm}}^{2}$/V s at 4.2 K and 3.1\ifmmode\times\else\texttimes\fi{}${10}^{5}$ ${\mathrm{cm}}^{2}$/s at 1.5 K for an electron density of 7.5\ifmmode\times\else\texttimes\fi{}${10}^{11}$ ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}2}$ (for typical choices of parameters: 10-nm spacer and 2\ifmmode\times\else\texttimes\fi{}${10}^{18}$ ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}3}$ doping). Peak mobility values of 5.0\ifmmode\times\else\texttimes\fi{}${10}^{5}$ ${\mathrm{cm}}^{2}$/V s at 4.2 K and 7.6\ifmmode\times\else\texttimes\fi{}${10}^{5}$ ${\mathrm{cm}}^{2}$/s at 1.5 K are possible for wider spacer layer widths, with subsequently lower channel electron densities. The effects of surface roughness scattering, as well as other scatterers are discussed. These processes can be a mechanism to explain the difference between the above ideal mobility and the reported experimental data.