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
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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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