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A technique is described for isolating effects of finite spin diffusion lengths on the magnetoresistance (MR) of magnetic multilayers at low temperatures by measuring the MR with the current perpendicular to the layer planes. Applying this technique to relatively concentrated Co/AgMn and Co/CuMn multilayers yields large effects, generally consistent with expectations from a theory by Valet and Fert [Phys. Rev. B 48, 7099 (1993)]. However, some complicating issues still need to be resolved.