Question

If a certain number is divided by the sum of its two digits, the quotient is $6$ and remainder is $3$. If the digits are interchanged and the resulting number is divided by the sum of the digits, then the quotient is $4$ and the remainder is $9$. Find the number.

• A. $50$
• B. $75$
• C. $65$
• D. $35$

Answer 1

Answer: (B)

$75$

Let the two digit number be $10x+y$ and number obtained by interchanging the digit $=10y+x$
As per the first condition,
$10x+y=6(x+y)+3$
$4x-5y=3...\left(i\right)$
As per the second condition,

$10y+x=4(x+y)+9$
$-3x+6y=9...\left(ii\right)$
Multiplying equation $\left(i\right)$ by $3$ and $\left(ii\right)$ by $4$ and adding them, we get
$12x-15y=9$
$-12x+24y=36$
------------------------
$9y=45$
$y=5$
Substitute $y=5$ in equation $\left(i\right)$
$4x-5\times 5=3$
$4x=28$
$x=7$
So, the number $=10x+y=10\times 7+5=75$