 Kevin, Alison, and Fred work part-time. Kevin works every second day. Alison works eve third day. Fred works every fourth day. Today they all work together. When will they work together again? Explain how you know by repeating the question, key info, how to find the solution, the calculations, and the answer. you can use factor trees, or factor rainbows

**Question Restatement:**
Kevin, Alison, and Fred work part-time. Kevin works every second day, Alison works every third day, and Fred works every fourth day. Today, all three are working together. The goal is to find out when they will all work together again.

**Key Information:**
1. Kevin works every 2nd day.
2. Alison works every 3rd day.
3. Fred works every 4th day.
4. Today they all work together, meaning we need to find the next time all three schedules align.

**How to Solve:**
To find out when they will all work together again, we need to determine the Least Common Multiple (LCM) of the numbers 2 (Kevin's cycle), 3 (Alison's cycle), and 4 (Fred's cycle). The LCM will tell us the number of days after which their schedules will align again.

**Finding the LCM:**
There are two methods we can use to find the LCM: factor trees and factor rainbows. Here, I will use both methods.

### Factor Trees:

1. **Kevin (2 days):**  
   2 is a prime number, so its prime factorization is simply 2.  
   - 2 = 2

2. **Alison (3 days):**  
   3 is a prime number, so its prime factorization is simply 3.  
   - 3 = 3

3. **Fred (4 days):**  
   4 is not a prime number, so we break it down into its prime factors:
   - 4 = 2 × 2

Now, we look at the prime factors of all the numbers:
- Kevin's cycle (2 days): 2
- Alison's cycle (3 days): 3
- Fred's cycle (4 days): 2 × 2

To find the LCM, we take the highest power of each prime factor:
- The highest power of 2 is \(2^2\) (from Fred's cycle).
- The highest power of 3 is \(3^1\) (from Alison's cycle).

So, the LCM = \(2^2 \times 3 = 4 \times 3 = 12\).

### Factor Rainbows:
Draw a rainbow connecting all the common multiples of 2, 3, and 4 until we reach the LCM:
- Multiples of 2: 2, 4, 6, 8, 10, 12...
- Multiples of 3: 3, 6, 9, 12...
- Multiples of 4: 4, 8, 12...

The first common multiple is 12.

**Final Answer:**
Kevin, Alison, and Fred will all work together again in 12 days. We found this by calculating the Least Common Multiple (LCM) of their work schedules (2, 3, and 4), which is 12.

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Would you like more details or have any questions about the process?  

Here are some related questions:
1. How do you find the LCM using division by prime numbers?
2. What is the relationship between the Greatest Common Divisor (GCD) and LCM?
3. How would the solution change if one person worked every 5th day?
4. What is the LCM of three numbers that have no common factors?
5. Why is the LCM useful in real-life scheduling problems?

**Tip:** The LCM is helpful for syncing repeated events, like schedules, as it shows the next time all events coincide.

Kevin, Alison, and Fred work part-time. Kevin works every second day. Alison works every third day. Fred works every fourth day. Today they all work together. When will they work together again? Explain how you know by repeating the question, key info, how to find the solution, the calculations, and the answer. You can use factor trees or factor rainbows.

Discover how to calculate the next time Kevin, Alison, and Fred will work together using Least Common Multiple (LCM). This problem breaks down their work schedules—Kevin works every 2 days, Alison every 3, and Fred every 4. The solution explains factor trees and factor rainbows to find the LCM, making it perfect for students in Grades 5-7.

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