Product of digits of a 2-digit number is 18. If we add 63 to the number, the new number obtained is a number formed by interchange of the digits. Find the number.

Let the unit's digit of the number be $$y$$ and ten's digit be $$x$$

=> Number = $$10x + y$$

Product of digits = $$x y = 18$$ --------------(i)

According to question, => $$10x + y + 63 = 10y + x$$

=> $$9y - 9x = 63$$

=> $$y - x = \frac{63}{9} = 7$$ --------------(ii)

Substituting value of $$y$$ from equation (ii) in (i), we get : 

=> $$x (7 + x) = 18$$

=> $$x^2 + 7x - 18 = 0$$

=> $$x^2 + 9x - 2x - 18 = 0$$

=> $$x(x + 9) - 2(x + 9) = 0$$

=> $$x = 2 , -9$$

Since $$x$$ is a digit and can't be negative, => $$x = 2$$

Substituting it in equation (ii), => $$y = 7 + 2 = 9$$

$$\therefore$$ Number = 29