Question

The sum of a two-digit number and the number obtained by reversing the order of its digits is $165$. If the digits differ by $3$, find the number.

Answer 1

Let the digits at units and tens place of the given number be x and y respectively. Then,
Number $=10y+x$ (i)

Number obtained by reversing the order of the digits $=10x+y$

According to the given conditions, we have
$(10y+x)+(10x+y)=165$
and, $x-y=3$ or, $y-x=3$

$\Rightarrow 11x+11y=165$
and, $x-y=3$ or, $y-x=3$

$\Rightarrow x+y=15$
and, $x-y=3$ or, $y-x=3$

Thus, we obtain the following systems of linear equations.
(i) $x+y=15$ .....(1)
$x-y=3$ ......(2)
(ii) $x+y=15$ .....(3)
$y-x=3$ ...... (4)

Adding equation (1) and (2), we get

$2x=18\Rightarrow x=9$

Putting in (1), we get

$y=15-9=6$
$x=9,y=6$

Adding equation (3) and (4), we get
$2y=18\Rightarrow y=9$

Putting in (3), we get

$x=15-9=6$

$x=6,y=9$

Substituting the values of x and y in equation (i), we have
Number $=69$ or $96$.