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If I get multiple BOA BBR cards would I need to leave a balance every single month on every single one of them?
If one wants the rewards, of course.
As Wave Rider pointed out, if you want the quarterly reward for each card, then each card needs to cycle with a balance. Now, for whatever reason one would have multiple BBRs but only wants the reward for one of them, then it is not necessary for all BBR cards to have the balance for just the one card to get the reward. But again, if you have two BBR and you want two quarterly rewards, both cards will have to cycle with a balance for all three months during that quarter.