What is the most important Federalist Paper and why?

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Federalist Paper 10 is often thought of as the most important, and most famous, federalist paper. Written by Madison, it deals with the problem of factions. A common fear for the new government was that small groups, or factions, would compromise the integrity and stability of the new government. Madison outlines the plan for a democracy that allows a vote per person, but also points towards the use of a republic, whereby citizens vote for delegates to make decisions for them (i.e., our modern Congress). He calls this a representative democracy, which we now have. See 1st quotation below. He sees this as a way to protect from minority factions taking over the government but also as way to protect minority opionion rights. "Fit people" as he called them would make decisions but would also have to negotiate, so that no one faction, or corrupt politician, could control the government. 

Madison favors a large republic and argues against individual states existing independently. He believes a greater republic will give a more diverse base to aquire thefort people who rule. He also outlines how factions will have less influence in a larger segment of people [see second quotation].

(1) A pure democracy ... can admit no cure for the mischiefs of faction. A common passion or interest will be felt by a majority, and there is nothing to check the inducements to sacrifice the weaker party. Hence it is, that democracies have ever been found incompatible with personal security or the rights of property; and have, in general, been as short in their lives as they have been violent in their deaths".

(2) In the first place, it is to be remarked that, however small the republic may be, the representatives must be raised to a certain number, in order to guard against the cabals of a few; and that, however large it may be, they must be limited to a certain number, in order to guard against the confusion of a multitude. Hence, the number of representatives in the two cases not being in proportion to that of the two constituents, and being proportionally greater in the small republic, it follows that, if the proportion of fit characters be not less in the large than in the small republic, the former will present a greater option, and consequently a greater probability of a fit choice.