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The conjugate seesaw says that the stronger the acid, the weaker that acid's conjugate base. The stronger the base, the weaker that base's conjugate acid. But here's why... We know that Ka * Kb =Kw, and we know that Kw is a the ionization constant of water= 1* 10^-14. Therefore if we have a large Ka, meaning that we have a strong base that dissociates a lot or even completely, then Kb must be small because Ka times Kb is a constant (1*10^-14). The same logic holds with a strong base.
Since Kw is a constant value, the strengths of acids and their conjugate bases (as well as bases and their conjugate acids) are complimentary. That would make sense if you think about it: the stronger an acid/base, the more likely it will stay disassociated and thus its conjugate base will be weaker and less likely to go back to the reactant form.
Basically, if there is an acid base reaction, the base (on the reactants side) will have a conjugate acid (on the products side), and vice versa. Since this is the case, it may be necessary to convert Ka to Kb (or the other way around) depending on what way you are looking at the reaction.
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