Why does Madison conclude that a pure democracy cannot solve the problems of faction but that a republic can solve them?

Madison concludes that a pure democracy cannot effectively address the issues of faction because it is prone to the tyranny of the majority, where the majority can impose its will on minorities, leading to instability and conflict. In contrast, a republic, with its system of elected representatives and a larger sphere of governance, can mitigate the effects of faction by diluting their influence and promoting a more deliberative process. This structure allows for a greater diversity of interests and opinions, making it more difficult for any single faction to dominate. Thus, a republic is better equipped to balance competing interests and promote the common good.