The digits of a positive integer having three dirgits are in A.P. and their sum is 15. If the number obtained by reversing the digits is 594 less than the original number, then the number is

Options

A. 352
B. 652
C. 852
D. none

Answer

To solve the problem step by step, we will denote the three digits of the number as \( a - d \), \( a \), and \( a + d \), where \( a \) is the middle digit and \( d \) is the common difference of the arithmetic progression (A.P.).

### Step 1: Set up the equations based on the problem statement.

1. **Sum of the digits**:
\[ (a - d) + a + (a + d) = 15 \]
Simplifying this gives:
\[ 3a = 15 \implies a = 5 \]

**Hint**: Remember that the sum of the digits in an A.P. can be simplified by combining like terms.

### Step 2: Substitute \( a \) back into the digits.

Now that we have \( a = 5 \), the digits can be expressed as:
- First digit: \( 5 - d \)
- Second digit: \( 5 \)
- Third digit: \( 5 + d \)

### Step 3: Write the original number and the reversed number.

The original number \( N \) can be expressed as:
\[ N = 100(5 - d) + 10(5) + (5 + d) = 500 - 100d + 50 + 5 + d = 555 - 99d \]

The reversed number \( R \) can be expressed as:
\[ R = 100(5 + d) + 10(5) + (5 - d) = 500 + 100d + 50 + 5 - d = 555 + 99d \]

### Step 4: Set up the equation based on the difference between the numbers.

According to the problem, the reversed number is 594 less than the original number:
\[ R = N - 594 \]
Substituting the expressions for \( N \) and \( R \):
\[ 555 + 99d = (555 - 99d) - 594 \]

### Step 5: Solve for \( d \).

Now, simplify the equation:
\[ 555 + 99d = 555 - 99d - 594 \]
Combine like terms:
\[ 555 + 99d + 99d = 555 - 594 \]
This simplifies to:
\[ 198d = -39 \]
Dividing both sides by 198 gives:
\[ d = -\frac{39}{198} = -\frac{1}{5} \]

### Step 6: Find the digits.

Now substituting \( d \) back into the expressions for the digits:
- First digit: \( 5 - d = 5 + \frac{1}{5} = 5.2 \) (not valid)
- Second digit: \( 5 \)
- Third digit: \( 5 + d = 5 - \frac{1}{5} = 4.8 \) (not valid)

Since \( d \) must be an integer, we need to re-evaluate our steps.

### Step 7: Correct the approach.

Let's assume \( d \) must be a whole number. Since \( a = 5 \), we can try possible values for \( d \) that keep \( 5 - d \) and \( 5 + d \) as digits (0-9).

### Step 8: Testing values for \( d \).

1. If \( d = 1 \):
- Digits: \( 4, 5, 6 \) → Number: 456
- Reversed: 654
- Difference: \( 654 - 456 = 198 \) (not valid)

2. If \( d = 2 \):
- Digits: \( 3, 5, 7 \) → Number: 357
- Reversed: 753
- Difference: \( 753 - 357 = 396 \) (not valid)

3. If \( d = 3 \):
- Digits: \( 2, 5, 8 \) → Number: 258
- Reversed: 852
- Difference: \( 852 - 258 = 594 \) (valid)

### Conclusion

The original number is **258**.