We have performed an ultrahigh-resolution angle-resolved photoemission spectroscopy study of slightly overdoped (Bi,Pb)2Sr2CuO6 to elucidate the origin of the pseudogap. By using a newly developed xenon-plasma light source, we determined the comprehensive momentum and temperature dependencies of the superconducting gap and the pseudogap. We found that the antinodal pseudogap persists far above the superconducting transition temperature and is smoothly connected to the nodal gap. The characteristic temperature of the pseudogap scales well with the superconducting gap size irrespective of the momentum location. The present experimental results point to the pairing origin of the pseudogap.