Question

In a two digit number, the units digit exceeds its tens digit by $2$. t product of the given number and the sum of its digits is equal to $144$. Find the number.

Answer 1

Let the units digit be $X$

And tens digits be $Y$

Number$=10Y+X$

Units digit exceeds tens digit by 2

Hence $X-Y=2$

$X=2+Y⟹\left(i\right)$

And $(10Y+X)(X+Y)=144$

$10XY+10Y^2+X^2+XY=144$

$11XY+10Y^2+X^2=144⟹\left(ii\right)$

Substitute $X$ from $\left(i\right)$ in $\left(ii\right)$ and get

$11(2+Y)Y+10Y^2+{(2+Y)}^2=144$

$22Y+11Y^2+10Y^2+4+4Y+Y^2=144$

$22X+11X^2+40+40X+10X^2=144$

$22Y^2+26Y-140=0$

$11Y^2+13Y-70=0$

$11Y^2+35Y-22Y-70=0$

$Y(11Y+35)-2(11Y+35)=0$

$(y-2)(11Y+35)=0$

Now either $(Y-2)=0$

$Y=2$

Or $11Y+35=0$

$Y=\frac{-35}{11}$

Since $Y$ is non negative

Hence $Y=2$ and $X=2+Y=2+2=4$

Hence required number =4