Question : If a certain number of two digits is divided by the sum of its digits, the quotient is 6 and the remainder is 3. If the digits are reversed and the resulting number is divided by the sum of the digits, the quotient is 4 and the remainder is 9. The sum of the digits of the number is:
Option 1: 6
Option 2: 9
Option 3: 12
Option 4: 4
Correct Answer: 12
Solution :
Let the number be $10x + y$.
Dividend = Divisor × Quotient + Remainder
$10x + y = 6(x + y) + 3$
⇒ $10x + y = 6x + 6y + 3$
⇒ $10x – 6x + y – 6y = 3$
⇒ $4x – 5y = 3$ (equation 1)
After interchanging the digits, we have:
$10y + x = 4 (x + y) + 9$
⇒ $10y + x = 4x + 4y + 9$
⇒ $6y – 3x = 9$
⇒ $2y – x = 3$ (equation 2)
From equation 1 and 2, we get:
$4x – 5y = 3$
$2y – x = 3$ (multiplying by 4)
Solving the equation, we get $y = 5$.
Putting the value of $y$ in equation (2),
$2×5 – x = 3$
⇒ $x = 10 – 3 = 7$
So, the sum of digits = $(x + y)=7 + 5 =12$
Hence, the correct answer is 12.
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