Question

A number consist of two digits. When it is divided by the sum of its digit, the quotient is $6$ with no reminder. When the number is diminished by $9$, the digits are reversed. Find the number.

Answer 1

We know that Dividend$=$Divisor$\times$Quotient$+$Remainder

Let the tens and the units digits of the required number be $x$ and $y$,Respectively.

Required number$=10x+y$

$10x+y=(x+y)\times 6+0$

$\Rightarrow 10x-6x+y-6y=0$

$\Rightarrow 4x-5y=0$ ......$\left(1\right)$

Number obtained on reversing its digits$=10y+x$

$\therefore 10x+y-9=10y+x$

$\Rightarrow 9x-9y=9$ or $x-y=1$ ........$\left(2\right)$

On multiplying $\left(2\right)$ by $5$, we get

$5x-5y=5$ .........$\left(3\right)$

On subtracting $\left(1\right)$ from $\left(3\right)$,we get

$x=5$

On substituting $x=5$ in $\left(1\right)$, we get

$4\times 5-5y=0$

$\Rightarrow 20-5y=0$

$\Rightarrow y=4$

$\therefore$The number $=10x+y=10\times 5+4=50+4=54$

Hence the required number is $54$