The digits of a three digit number N are in A.P. If sum of the digit is 15 and the number obtained by re-versing the digits of the number is 594 less than the original number, then `(1000)/(N-252)` is equal to

Options

A. `5//6`
B. `5//3`
C. `0.06`
D. `0.03`

Answer

To solve the problem step by step, we will denote the three-digit number \( N \) as having digits \( a - d \), \( a \), 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. The digits are in A.P., so we can express them as:
- Hundreds place: \( a + d \)
- Tens place: \( a \)
- Units place: \( a - d \)

2. The sum of the digits is given as 15:
\[ (a + d) + a + (a - d) = 15 \]
Simplifying this gives:
\[ 3a = 15 \implies a = 5 \]

### Step 2: Find the value of \( d \)
3. Now, substituting \( a = 5 \) into the equation for the digits:
\[ (5 + d) + 5 + (5 - d) = 15 \]
This simplifies to:
\[ 15 = 15 \quad \text{(which is always true)} \]

4. Next, we use the information that the number obtained by reversing the digits is 594 less than the original number:
- Original number \( N = 100(5 + d) + 10(5) + (5 - d) \)
- Reversed number \( R = 100(5 - d) + 10(5) + (5 + d) \)

### Step 3: Set up the equation for the difference
5. The equation for the difference between the original number and the reversed number is:
\[ N - R = 594 \]
Substituting for \( N \) and \( R \):
\[ [100(5 + d) + 10(5) + (5 - d)] - [100(5 - d) + 10(5) + (5 + d)] = 594 \]

### Step 4: Simplify the equation
6. Simplifying the left side:
\[ (500 + 100d + 50 + 5 - d) - (500 - 100d + 50 + 5 + d) = 594 \]
This simplifies to:
\[ (100d - d + 100d + d) = 594 \implies 198d = 594 \]

### Step 5: Solve for \( d \)
7. Dividing both sides by 198:
\[ d = \frac{594}{198} = 3 \]

### Step 6: Find the digits of \( N \)
8. Now we can find the digits of \( N \):
- Hundreds place: \( 5 + 3 = 8 \)
- Tens place: \( 5 \)
- Units place: \( 5 - 3 = 2 \)

Thus, the number \( N \) is \( 852 \).

### Step 7: Calculate \( \frac{1000}{N - 252} \)
9. Now we need to calculate:
\[ N - 252 = 852 - 252 = 600 \]
Therefore:
\[ \frac{1000}{600} = \frac{10}{6} = \frac{5}{3} \]

### Final Answer
The value of \( \frac{1000}{N - 252} \) is \( \frac{5}{3} \).