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In the units place $$A + A$$ ends in 8, so A is 4 or 9. If $$A = 9$$, then $$9 + 9 = 18$$ gives 8 carry 1, and the tens place gives $$9 + 9 + 1 = 19$$, which shows A = 9 with carry 1. Then $$B + B + 1 = 3$$ gives $$B = 1$$, and indeed $$199 + 199 = 398$$.
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