If $$ \begin{array}{l} ab \overline{ ) 252 ( } ba \\ \ \ \ \ \ \underline{ 24 } \\ \ \ \ \ \ \ \ 12 \\ \ \ \ \ \ \ \underline{ 12 } \\ \ \ \ \ \ \ \ \times \end{array} $$ the values of $a$ and $b$ are

### Concept & Logic This problem relies on reverse-engineering the steps of long division, recognizing that each subtraction step represents the divisor multiplied by a single digit of the quotient. ### Step-by-Step Solution * **Given:** * Divisor: $ab$ (a 2-digit number) * Quotient: $ba$ (a 2-digit number) * Dividend: $252$ * **Calculation / Deduction:** 1. In the first step of the long division, the divisor ($ab$) is multiplied by the first digit of the quotient ($b$) to get $24$. $$ab \times b = 24$$ 2. After subtracting $24$ from $25$, the remainder is $1$, and the next digit ($2$) is brought down to form $12$. 3. In the second step, the divisor ($ab$) is multiplied by the second digit of the quotient ($a$) to yield $12$. $$ab \times a = 12$$ 4. Since $a$ and $b$ are individual digits, $ab$ is a two-digit number. The only two-digit number that can multiply by an integer to yield $12$ is $12$ itself. Therefore, $ab = 12$, which means $a = 1$ and $b = 2$. 5. Let's verify this against the first equation: If $a = 1$ and $b = 2$, does $12 \times 2 = 24$? Yes. ### Exam Strategy & Shortcut Option Elimination is the fastest method here. Test Option (a) where $a=1, b=2$: Divisor is $12$, quotient is $21$. First step of division: $12 \times 2 = 24$. Second step: $12 \times 1 = 12$. This perfectly matches the visual representation in the question. You can mark (a) and move on in 15 seconds. ### Common Pitfall Students often misinterpret the quotient variables. The quotient is $ba$, meaning the *first* division step uses the tens digit $b$, and the *second* step uses the units digit $a$. Reversing these would lead to confusion and an inability to satisfy $ab \times a = 12$. ### Final Answer Therefore, the correct answer is 1, 2.