Question

A two digit number is such that the product of its digits is $12$.When $36$ is added to this number, the digits are interchanged.Find the number

Answer 1

Let the units digit of the two-digit number be $x$

Since the product of its digits is $12$, its ten's digit is $\frac{12}{x}$

$\therefore$ the number is $10\left(\frac{12}{x}\right)+x$

On interchanging the digits, the number is $10x+\frac{12}{x}$

Given:$\frac{120}{x}+x+36=10x+\frac{12}{x}$

$\Rightarrow \frac{120-12}{x}+x-10x+36=0$

$\Rightarrow \frac{108}{x}-9x+36=0$

$\Rightarrow \frac{12}{x}-x+4=0$

$\Rightarrow x^2-4x-12=0$

$\Rightarrow x^2-6x+2x-12=0$

$\Rightarrow x(x-6)+2(x-6)=0$

$\therefore x=6$ or $x=-2$ but $x$ being a digit of a number cannot be negative.

$\therefore x=6$

$\therefore$ Units's digit$=6$ and ten's digit$=\frac{12}{6}=2$

Hence the required number is $26$