\[ \textbf{Step 1: Let the two-digit number be } 10x + y, \] where \(x\) is the tens digit and \(y\) is the units digit. \[ \textbf{Step 2: Condition 1: Product of digits } \Rightarrow x \cdot y = 12. \] \[ \textbf{Step 3: Condition 2: Adding 36 reverses the digits } \Rightarrow 10x + y + 36 = 10y + x. \] \[ \textbf{Step 4: Simplify equation 2:} \] \[ 10x + y + 36 = 10y + x \quad \Rightarrow \quad 9x - 9y = -36 \quad \Rightarrow \quad x - y = -4. \] \[ \therefore x = y - 4. \] \[ \textbf{Step 5: Use condition 1: } x \cdot y = 12. \] \[ (y - 4)\cdot y = 12 \quad \Rightarrow \quad y^2 - 4y - 12 = 0. \] \[ \textbf{Step 6: Solve quadratic: } y = \frac{4 \pm \sqrt{16 + 48}}{2} = \frac{4 \pm 8}{2}. \] \[ y = 6 \quad \text{or} \quad y = -2 \ (\text{not possible since digit}). \] \[ \textbf{Step 7: If } y = 6, \quad x = y - 4 = 2. \] \[ \textbf{Final Answer: } \boxed{26} \]