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No. If you have a strong acid, that means that it is totally deprotonated in solution. A strong base would be completely protonated in solution-- so the conjugate of either would have to be weak or else, by definition, the conjugate acid or base would completely ionize again to form the original acid/base.
They have conjugate acids/bases, they are just extremely weak. Acid and conjugate base strength are inversely proportional. HBr is a strong acid, so its conjugate base, Br-, is so weak that it is an ineffective base.
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