Question

A two digit number is such that the product of the digits is $35$, when $18$ is added to the number, the digits interchange their places. Find the number.

Answer 1

Solution:-

Let the two digit no. be $xy$.

Then its decimal expansion becomes $10\times x+y$

When digits are interchanged, the number is $yx$.

Then its decimal expansion becomes $10\times y+x$

$x\times y=35\left\{Given\right\}$

$\Rightarrow x=\frac{35}{y}⟶\left(i\right)$

Given that when 18 is added to the number, the digits are interchanged

$\therefore (10x+y)+18=(10y+x)$

$\Rightarrow 9x-9y+18=0$

$\Rightarrow 9(x-y+2)=0$

$\Rightarrow x-y+2=0$

$\Rightarrow \frac{35}{y}-y+2=0\left\{\text{from}\left(i\right)\right\}$

$\Rightarrow 35-y^2+2y=0$

$\Rightarrow y^2-2y-35=0$

$\Rightarrow (y+5)(y-7)=0$

$\because y\ne -5$

$\therefore y=7$

Substituting the value of $y$ in $\left(i\right)$, we get

$x=\frac{35}{7}=5$

Hence the two digit no. is $57$.