A three digit number which on being subtracted from another three digit number consisting of the same digits in reverse order gives 594. The minimum possible sum of all the three digits of this number is :

Options

A. 808
B. 102
C. 6
D. can't be determined

Answer

To solve the problem step by step, we need to find a three-digit number \( xyz \) such that when it is subtracted from its reverse \( zyx \), the result is 594. We also want to minimize the sum of the digits \( x + y + z \).

### Step 1: Define the three-digit number and its reverse
Let the three-digit number be represented as:
\[ N = 100x + 10y + z \]
where \( x, y, z \) are the digits of the number. The reverse of this number will be:
\[ R = 100z + 10y + x \]

### Step 2: Set up the equation based on the problem statement
According to the problem, the difference between the reverse and the original number is 594:
\[ R - N = 594 \]
Substituting the expressions for \( R \) and \( N \):
\[ (100z + 10y + x) - (100x + 10y + z) = 594 \]

### Step 3: Simplify the equation
Now, simplify the equation:
\[ 100z + 10y + x - 100x - 10y - z = 594 \]
This simplifies to:
\[ 99z - 99x = 594 \]
Factoring out 99 gives:
\[ 99(z - x) = 594 \]
Dividing both sides by 99:
\[ z - x = 6 \]

### Step 4: Relate the digits
From the equation \( z - x = 6 \), we can express \( z \) in terms of \( x \):
\[ z = x + 6 \]

### Step 5: Determine the valid range for \( x \) and \( z \)
Since \( x \) and \( z \) are digits (0-9), and \( z \) must also be a digit:
- The maximum value for \( x \) can be 3 (because \( z \) must be less than or equal to 9).
- Therefore, \( x \) can take values 0, 1, 2, or 3.

### Step 6: Calculate possible values for \( z \) and \( y \)
1. If \( x = 3 \), then \( z = 9 \) and \( y \) can be 0 (to minimize the sum).
- Sum: \( 3 + 0 + 9 = 12 \)

2. If \( x = 2 \), then \( z = 8 \) and \( y \) can be 0.
- Sum: \( 2 + 0 + 8 = 10 \)

3. If \( x = 1 \), then \( z = 7 \) and \( y \) can be 0.
- Sum: \( 1 + 0 + 7 = 8 \)

4. If \( x = 0 \), then \( z = 6 \) and \( y \) can be 0.
- Sum: \( 0 + 0 + 6 = 6 \)

### Step 7: Identify the minimum sum
The minimum sum of the digits occurs when:
- \( x = 0 \), \( y = 0 \), \( z = 6 \)
- The sum is \( 6 \).

### Conclusion
The minimum possible sum of all the three digits of this number is:
\[ \boxed{6} \]