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The stability of the conjugate base is attributed to resonance and electron withdrawing. You can think about strength in two ways: proton donating as previous comments have said, or because the more stable a conjugate base is, the less likely it will actually act as a base and accept a proton, forming the molecule again (reversing the reaction). For example, since Cl- is a weak conjugate base of HCl, it won't readily accept protons to from HCl; this is why HCl completely ionizes.
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