Question

The product of the digits of a two-digit number is one-third that number. If we add $18$ to the required number, we get a number consisting of the same digits written in the reverse order. Find the number.

Answer 1

Let $ab$ be the two digit number.

Product of digits $=\frac{1}{3}\left(number\right)$

$ab=\frac{1}{3}(10a+b)$

$3ab=10a+b$

$number+18=\text{number with same digits in reverse order}$

$10a+b+18=10b+a$

$9(b-a)=18$

$b-a=2$

$b=a+2$

$3a(a+2)=10a+a+2$

$3a^2-5a-2=0$

$(3a+1)(a-2)=0$

$a=2(\because a\ne \frac{-1}{3})$

$b=4$

$\therefore 24$ is required number.