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We study distributed consensus problems of multi-agent systems on directed networks and subject to quantized information flow. For the communication among component agents, particular attention is given to the gossip type, which models their asynchronous behavior; for quantization effect, each agent's state is abstracted to be an integer. The central question investigated is how to design distributed algorithms and what connectivity of networks that together lead to consensus. This investigation is carried out for both general consensus and average consensus; for each case, a class of algorithms is proposed, under which a necessary and sufficient graphical condition is derived to guarantee the corresponding consensus. In particular, the obtained graphical condition ensuring average consensus is weaker than those in the literature for either real-valued or quantized states, in the sense that it does not require symmetric or balanced network topologies.