Twice the Product

Number Theory Level pending

2 × A × B = A B ‾ 2 \times A \times B = \overline{AB}

A A and B B are non-zero digits. What is A B ‾ \overline{AB} ?


The answer is 36.

Chung Kevin by Chung Kevin , 46, Australia

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1 solution

2 A B = A B ‾ ⟹ 2 A B = 10 A + B 2 A B − 10 A = B ⟹ A = B 2 B − 10 \begin{aligned} 2AB & = \overline{AB} \\ \implies 2AB & = 10A+B \\ 2AB - 10A & = B \\ \implies A & = \frac B{2B-10} \end{aligned}

We note that for A A to be a positive integer, B ≥ 6 B \ge 6 . For B = 6 B=6 , ⟹ A = 3 \implies A = 3 . There is no integer solution for B = 7 , 8 , 9 B=7,8,9 . Therefore A B ‾ = 36 \overline{AB} = \boxed{36} .

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