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We investigate the spin-dependent conductance of the Co/Cu multilayers in the current-perpendicular-tothe- plane (CPP) geometry. Using a realistic tight-binding model for the electronic band structure of the multilayer, and introducing disorder in the on-site atomic energies we calculate the conductance and giant magnetoresistance (GMR) within the quantum-mechanical linear response theory by performing averaging over random disorder configurations. By varying the thickness and the number of individual layers and the degree of disorder in the multilayers, we analyze factors influencing the CPP GMR. In particular, we show the importance of the thickness-dependent interface resistance, which depends on the mean free path and consequently lies beyond the two-current series-resistor model.