On a stretch of road, bus stops are located at interval of 200 m and phone booths are located at interval of 240  m. If a phone booth is located at a bus stop then
(a) How far along the road from the first phone booth would you be able to find the next phone booth located at a bus stop?
(b) How many phone booths and bus stops would you pass by before you find a phone booth at a bus stop for the second time?

To solve this problem, we need to determine the distance along the road at which a bus stop and a phone booth coincide, which means we need to find the **Least Common Multiple (LCM)** of the distances between the bus stops and phone booths.

### (a) Distance to the next phone booth located at a bus stop:

- The bus stops are located at intervals of 200 meters.
- The phone booths are located at intervals of 240 meters.

To find the first distance where both coincide, we calculate the LCM of 200 meters and 240 meters.

#### Steps to find the LCM:
1. **Prime factorization** of the distances:
   - \( 200 = 2^3 \times 5^2 \)
   - \( 240 = 2^4 \times 3 \times 5 \)

2. **LCM** is found by taking the highest power of each prime factor:
   - For \( 2 \): \( \text{highest power} = 2^4 \)
   - For \( 3 \): \( \text{highest power} = 3^1 \)
   - For \( 5 \): \( \text{highest power} = 5^2 \)

\[
\text{LCM} = 2^4 \times 3 \times 5^2 = 16 \times 3 \times 25 = 1200 \text{ meters}
\]

So, the next phone booth located at a bus stop would be 1200 meters from the first phone booth.

### (b) Number of phone booths and bus stops passed before the next coincidence:

- **Number of bus stops passed**: 
  \[
  \text{Number of bus stops} = \frac{1200 \text{ meters}}{200 \text{ meters/bus stop}} = 6 \text{ bus stops}
  \]

- **Number of phone booths passed**: 
  \[
  \text{Number of phone booths} = \frac{1200 \text{ meters}}{240 \text{ meters/phone booth}} = 5 \text{ phone booths}
  \]

Therefore, you would pass by **5 phone booths** and **6 bus stops** before finding a phone booth at a bus stop for the second time.

### Summary:
- **(a)** The distance to the next phone booth located at a bus stop is **1200 meters**.
- **(b)** You would pass by **5 phone booths** and **6 bus stops** before you find a phone booth at a bus stop for the second time.

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Would you like more details on the steps, or do you have any questions?

Here are some related questions:
1. How is the LCM used in real-world problems like this?
2. What would happen if the intervals were not multiples of each other?
3. How can you calculate the number of coincidences within a certain distance?
4. Can this problem be solved using GCD instead of LCM?
5. How does this problem change if the intervals were reduced?

**Tip:** Knowing how to calculate the LCM is very useful in problems involving repeated events or patterns, such as scheduling, distances, or synchronization tasks.

Explore how to determine the distance to the next phone booth located at a bus stop based on intervals of 200 meters for bus stops and 240 meters for phone booths. Understand the concept of Least Common Multiple (LCM) and its application in real-world distance problems. Suitable for high school students.

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