The dependences of the transport scattering time τt, quantum lifetime τq, and their ratio τt/τq on the density ne of the electron gas in modulation-doped single GaAs quantum wells with AlAs/GaAs short-period super-lattice barriers are investigated. The experimental dependences are explained in terms of electron scattering by remote ionized donors with an effective two-dimensional concentration nR* and background impurities with a three-dimensional concentration nB. An expression for nR*(ne) is obtained including the contribution of X-valley electrons localized in AlAs layers to the suppression of scattering by the random potential of remote donors. It is shown that the experimentally observed abrupt increase in τt and τq with an increase in ne above a certain critical value nec is related to a decrease in nR*. It is established that the drop in τt/τq observed for electron densities ne > nec occurs because scattering by the random potential of background impurities in this two-dimensional system with a decrease in nR* limits an increase in τt more considerably than an increase in τq.