nonlinearity can be used to convert an incoming single microwave photon into an outgoing n-photon Fock state in a different mode. In this process the Coulomb energy released by Cooper-pair tunneling is transferred to the outgoing Fock state, providing energy gain. The conversion can be made reflectionless (impedance-matched) so that all incoming photons are converted to n-photon states. With realistic parameters multiplication ratios n>2 can be reached. By cascading two to three such multiplication stages, the outgoing Fock-states can be sufficiently large to accurately discriminate them from vacuum with linear post-amplification and classical power measurement, implying that our scheme can be used as single-photon detector for itinerant microwave photons without dead time.