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If an acid is a strong acid, it can easily donate a proton (Ka is big, since products favored). Thus, when you look at it's conjugate base, the conjugate base would not likely accept a proton, as it wouldn't want the H+ back that it so readily gave up. Think of it this way. An acid would technically disassociate into a proton and its conjugate base. If it's a strong acid, the reaction would favor the formation of that proton and that conjugate base. Thus, if we were to flip the equation so that the conjugate base and proton are now the reactants, the reaction would not favor the formation of that strong acid; it would favor the reverse reaction (thus the conjugate base's Kb would be small, since reactants favored). This is why the conjugate of a strong acid is a weak base, and vice versa.
We also know that Ka x Kb = Kw, with Kw being a constant of 1.0X10^14. So according to this equation, increasing Ka would mean Kb needs to decrease and vice versa, so if you have a strong acid, you automatically get a weaker conjugate base. Hope this helps
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