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Strong acids and bases are just things you're going to have to memorize. Generally, strong bases are group 1 and group 2 metals with an OH group, meaning their conjugate ions are the metals. Strong acids are HBr, HCl, HI, HNO3, HClO4, HSO4, and their conjugate ions would just be those without the H.
If there is a strong acid, then the conjugate base will be weaker. Similarly if you are presented with a weak acid, then the conjugate base will be stronger. This trend is also true when evaluating the conjugate bases of weak/strong acids.
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