Question

A two-digit number is $3$ more than six times the sum of its digits. If $18$ is added to the number obtained by interchanging the digits, we get the original number. Find the number.

Answer 1

Let unit's place digit $=\alpha$, ten's digit $=\beta$

$\Rightarrow 10\beta +\alpha =3+6(\alpha +\beta )$

$\Rightarrow 4\beta -5\alpha =3$

Also $10\alpha +\beta +18=10\beta +\alpha$

$\Rightarrow 9\beta -9\alpha =18$

$\Rightarrow \beta -\alpha =2$

$\Rightarrow \beta =\alpha +2$

$\Rightarrow 4\alpha +8-5\alpha =3$

$\Rightarrow \alpha =5$

$\Rightarrow \beta =7$

$\Rightarrow$ Number $=75$.