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The capacity regions of two classes of multiple-access relay channels (MARCs) are characterized. The MARCs studied in this paper include a primitive relay component, which can help in relaying the messages through a non-interfering finite-capacity link to the decoder. The capacity regions for such MARCs are characterized when: (a) the decoders output is a physically degraded version of the relays output or, (b) the relay output can be obtained as a deterministic function of the decoder output and any one of the channel inputs. For both these classes of MARCs, it is shown that the cut-set outer bound is achievable. As a consequence of the result for case (b), the capacity of a class of state-dependent multiple-access channels with rate-limited state information at the decoder is also established.