Beth and Andy ended up making the same number of cookies for a bake sale at school, even though Beth made them in batches of 7 cookies and Andy made them in batches of 6 cookies. What is the smallest number of cookies each must have baked?

To determine the smallest number of cookies each must have baked, we need to find the least common multiple (LCM) of the two numbers: 7 (Beth's batch size) and 6 (Andy's batch size). The LCM is the smallest number that is a multiple of both 7 and 6.

### Step 1: Find the prime factorization of each number.
- **Prime factorization of 7:**
  - \(7 = 7^1\)
- **Prime factorization of 6:**
  - \(6 = 2 \times 3\)

### Step 2: Identify all the prime factors and their highest powers.
- The prime factors involved are 2, 3, and 7.
- Highest power of 2 is \(2^1\).
- Highest power of 3 is \(3^1\).
- Highest power of 7 is \(7^1\).

### Step 3: Multiply the highest powers of all prime factors.
- LCM = \(2^1 \times 3^1 \times 7^1 = 2 \times 3 \times 7 = 42\)

### Conclusion:
The smallest number of cookies each must have baked is **42**.

Would you like any additional details or have any questions?

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Here are five questions related to the problem:
1. How many batches did Beth and Andy each bake to reach 42 cookies?
2. What if Beth made cookies in batches of 8 and Andy in batches of 6—what would be the smallest number of cookies then?
3. Why is finding the LCM important in solving problems like these?
4. How would the LCM change if Beth made cookies in batches of 7 and Andy in batches of 9?
5. Can the concept of LCM be applied to situations with more than two different batch sizes?

**Tip:** The LCM is a crucial concept for determining the smallest common multiple when dealing with different quantities or batch sizes, ensuring that no amount is left out or added unnecessarily.

Learn how to find the smallest number of cookies Beth and Andy must have baked for a school bake sale. This problem involves calculating the Least Common Multiple (LCM) of batch sizes 7 and 6, ensuring no cookies are left out. Suitable for students in Grades 5-7.

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