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We consider reliable transmission of a discrete memoryless source over a cooperative relay broadcast channel, where both the relay and the destination terminals want to reconstruct the source; and over a relay channel, where only the destination terminal wishes to obtain a lossless reconstruction. We assume that both the relay and the destination have correlated side information. We find the necessary and sufficient conditions for a general cooperative relay broadcast channel, and for a physically degraded relay channel when the side information at the destination is a degraded version of the relay side information. Our achievability results are based on operational source-channel separation. We utilize source and channel codes that interact only by passing along decoded source codewords from one block to another.