Question

A number consists of two digits whose product is 18. If 27 is added to the number, the digits interchange their places. Find the number.

Answer 1

Let the ten’s digit be $$x$$.

Since the product of digits is 18, the unit’s digit is $$\frac{18}{x}$$. 

Original number $$= 10x + \frac{18}{x}$$. 

Number obtained by interchanging the digits $$= 10\left(\frac{18}{x}\right) + x = \frac{180}{x} + x$$. 

According to the question: $$\left(10x + \frac{18}{x}\right) + 27 = \frac{180}{x} + x$$

$$9x + 27 - \frac{162}{x} = 0$$ 

Dividing throughout by 9: $$x + 3 - \frac{18}{x} = 0$$

$$x^2 + 3x - 18 = 0$$ 

Factoring: $$(x + 6)(x - 3) = 0$$

$$x = -6 \quad \text{or} \quad x = 3$$ 

Since a digit cannot be negative, reject $$x = -6$$. 

Therefore, ten’s digit $$x = 3$$ and unit’s digit $$= \frac{18}{3} = 6$$. 

Hence, the required number is 36.