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This paper addresses lossy transmission of a common source over a broadcast channel when there is correlated side information at the receivers, with emphasis on the quadratic Gaussian and binary Hamming cases. A digital scheme that combines ideas from the lossless version of the problem, i.e., Slepian-Wolf coding over broadcast channels, and dirty paper coding, is presented and analyzed. This scheme uses layered coding where the common layer information is intended for both receivers and the refinement information is destined only for one receiver. For the quadratic Gaussian case, a quantity characterizing the combined quality of each receiver is identified in terms of channel and side information parameters. It is shown that it is more advantageous to send the refinement information to the receiver with ??better?? combined quality. In the case where all receivers have the same overall quality, the presented scheme becomes optimal. Unlike its lossless counterpart, however, the problem eludes a complete characterization.