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

Options

A. 6
B. 9
C. 12
D. 4

Answer

To solve the problem step by step, we will denote the two-digit number as \(10x + y\), where \(x\) is the tens digit and \(y\) is the units digit.

### Step 1: Set up the first equation
According to the problem, when the number \(10x + y\) is divided by the sum of its digits \(x + y\), the quotient is 6 and the remainder is 3. This can be expressed mathematically as:

\[ 10x + y = 6(x + y) + 3 \]

### Step 2: Simplify the first equation
Expanding the equation gives:

\[ 10x + y = 6x + 6y + 3 \]

Now, rearranging the terms:

\[ 10x + y - 6x - 6y = 3 \]

This simplifies to:

\[ 4x - 5y = 3 \quad \text{(Equation 1)} \]

### Step 3: Set up the second equation
Next, when the digits are reversed, the number becomes \(10y + x\). The problem states that when this reversed number is divided by the sum of its digits \(x + y\), the quotient is 4 and the remainder is 9. This can be expressed as:

\[ 10y + x = 4(x + y) + 9 \]

### Step 4: Simplify the second equation
Expanding this equation gives:

\[ 10y + x = 4x + 4y + 9 \]

Rearranging the terms:

\[ 10y + x - 4x - 4y = 9 \]

This simplifies to:

\[ -3x + 6y = 9 \quad \text{(Equation 2)} \]

### Step 5: Solve the system of equations
Now we have a system of equations:

1. \(4x - 5y = 3\) (Equation 1)
2. \(-3x + 6y = 9\) (Equation 2)

To eliminate \(x\), we can multiply Equation 1 by 3 and Equation 2 by 4:

\[ 3(4x - 5y) = 3(3) \implies 12x - 15y = 9 \quad \text{(Equation 3)} \]
\[ 4(-3x + 6y) = 4(9) \implies -12x + 24y = 36 \quad \text{(Equation 4)} \]

### Step 6: Add the equations
Now we add Equation 3 and Equation 4:

\[ (12x - 15y) + (-12x + 24y) = 9 + 36 \]

This simplifies to:

\[ 9y = 45 \]

### Step 7: Solve for \(y\)
Dividing both sides by 9 gives:

\[ y = 5 \]

### Step 8: Substitute \(y\) back to find \(x\)
Now we substitute \(y = 5\) back into Equation 1 to find \(x\):

\[ 4x - 5(5) = 3 \]
\[ 4x - 25 = 3 \]
\[ 4x = 28 \]
\[ x = 7 \]

### Step 9: Find the original number and the sum of its digits
The original number is:

\[ 10x + y = 10(7) + 5 = 75 \]

The sum of the digits is:

\[ x + y = 7 + 5 = 12 \]

### Final Answer
The sum of the digits of the number is \(12\).

---