Arpita wanted to go on a holiday with her friends on the long weekend in her brand new car which had covered less than 200 km. The odometer reading when she returned back home
after the weekend was exactly reverse of the reading at the time that she started from home. The total distance covered was divisible by 77. What is the farthest distance that the holiday
destination could have been from Arpita's home, if she took the same route either way? (to the nearest integer kilometer)

Let's denote the odometer reading at the time Arpita started from home as \( N \) and the reading when she returned as \( N' \), where \( N' \) is the reverse of \( N \). The total distance she covered during her trip is \( 2D \), where \( D \) is the distance from her home to the holiday destination. The problem also states that the total distance covered is divisible by 77, meaning \( 2D \) is divisible by 77.

### Step-by-step approach:

1. **Reversal condition**: 
   The odometer reading after her trip is the reverse of the initial reading. This means if her initial odometer reading is \( N \), then the final reading is \( N' \), the reverse of \( N \).

2. **Distance covered**:
   The total distance covered is \( N' - N \), which equals \( 2D \), the round-trip distance. Additionally, this total distance is divisible by 77.

   \[
   N' - N = 2D
   \]
   \[
   2D \text{ is divisible by } 77
   \]

3. **Determine values for \( N \) and \( N' \)**:
   The problem specifies that the initial odometer reading \( N \) was less than 200 km. We need to find the maximum possible value for \( D \) such that the total distance \( N' - N \) is divisible by 77.

Let’s now calculate the odometer values satisfying these conditions.The farthest distance that Arpita's holiday destination could have been from her home, assuming she took the same route both ways, is approximately **346 kilometers**.

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. How does reversing numbers work in mathematical problems?
2. Why is divisibility important in problems like this?
3. What would change if the distance was divisible by a different number, like 11?
4. How does modular arithmetic apply to problems involving divisibility?
5. How would the result differ if the starting odometer reading was above 200?

**Tip:** In problems where numbers are reversed, systematically checking small cases can often reveal patterns that solve the problem.

Arpita wanted to go on a holiday with her friends on the long weekend in her brand new car which had covered less than 200 km. The odometer reading when she returned back home after the weekend was exactly reverse of the reading at the time that she started from home. The total distance covered was divisible by 77. What is the farthest distance that the holiday destination could have been from Arpita's home, if she took the same route either way? (to the nearest integer kilometer)

Solve the problem where Arpita's odometer reading after a trip is the reverse of the starting reading, and the total distance covered is divisible by 77. This solution covers key number theory concepts and formulas, suitable for high school students (Grades 9-12).

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