As another clarification, all National Banks (and only National Banks) must have the word National in their name. Many have it incorporated as "First National Bank of XXX"
Some banks use the N.A. at the end and there are other variations such as the original Bank of America N.T. & S.A. (National Trust and Savings Association). It means they are chartered and examined by the Office of the Comptroller of the Currency.
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Integrity. With it, nothing else matters. Without it, nothing else matters.