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.
29
92
36
63
Solution
Let the unit's digit of the number be $y$ and ten's digit be $x$
$\Rightarrow$ Number = $10x + y$
Product of digits $= xy = 18~~~~~~~~~~----$(i)
According to question,
$\Rightarrow 10x + y + 63 = 10y + x$
$\Rightarrow 9y -9x = 63$
$\Rightarrow y - x = \dfrac{63}{9} = 7~~~~~~~~~~----$(ii)
Substituting value of $y$ from equation (ii), in (i), we get:
$\Rightarrow x(7 + x)=18$
$\Rightarrow x^2 + 7x - 18 = 0$
$\Rightarrow x^2 + 9x - 2x - 18$
$\Rightarrow x(x + 9) - 2(x + 9) = 0$
$\Rightarrow x = 2, -9$
Since $x$ is a digit and can't be negative, $\Rightarrow x = 2$
Substituting it in equation (ii), $\Rightarrow y = 7 + 2 = 9$
$\therefore$ Number = 29