Question

A two-digit number is such that the product of its digits is 12. If 36 is added to the number, the digits interchange their places. What is the number?

• A. 24
• B. 38
• C. 26
• D. 25

Hint:

When solving for two-digit numbers with conditions, use algebraic equations to express the relationships between the digits.

Answer 1

The Correct Option is C

Step 1: Define the digits.
Let the tens digit of the number be \( x \) and the ones digit be \( y \). Then, the number can be expressed as:
\[ 10x + y. \] We are given two conditions:
1. The product of the digits is 12, so:
\[ x \cdot y = 12. \] 2. If 36 is added to the number, the digits interchange their places, so:
\[ 10x + y + 36 = 10y + x. \]

Step 2: Solve the system of equations.
From the second equation:
\[ 10x + y + 36 = 10y + x $\Rightarrow$ 9x - 9y = -36 $\Rightarrow$ x - y = -4 \text{(Equation 1)}. \] From the first equation:
\[ x \cdot y = 12 \text{(Equation 2)}. \]

Step 3: Solve for \( x \) and \( y \).
From Equation 1:
\[ x = y - 4. \] Substitute this into Equation 2:
\[ (y - 4) \cdot y = 12 $\Rightarrow$ y^2 - 4y = 12 $\Rightarrow$ y^2 - 4y - 12 = 0. \] Solving this quadratic equation using the quadratic formula:
\[ y = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-12)}}{2(1)} = \frac{4 \pm \sqrt{16 + 48}}{2} = \frac{4 \pm \sqrt{64}}{2} = \frac{4 \pm 8}{2}. \] Thus, \( y = 6 \) or \( y = -2 \). Since \( y \) must be a positive digit, we take \( y = 6 \). Substituting \( y = 6 \) into \( x = y - 4 \), we get \( x = 2 \).

Step 4: Conclusion.
Thus, the number is \( 10x + y = 10(2) + 6 = 26 \), and the correct answer is (c).