The Digits

Let $x$ represent the tens digit and $y$ represent the units digit. Then the information given can be translated mathematically as $xy = x+y+39$ and $10y + x = 10x + y + 27$ . Subtracting $x + y$ from both sides of the second equation gives us $9y = 9x + 27$ . Dividing by $9$ yields $y = x + 3$ . Substituting $x+3$ for $y$ in the first equation gives us $x(x+3) = x + (x+3) + 39$ . Expanding and grouping like terms yields $x^2 + x - 42 = 0$ . Factoring this equation, we get $(x+7)(x-6) = 0$ . Since $x$ must be positive, we get that $x = 6$ . Since $y = x+3$ , $y = 9$ . So our number must be 69.