Cups are sold 6 to a package and plates are sold 8 to a package. If you want to have the same number of each item for a party, what is the least number of packages of cups you would need to buy?

To solve this problem, we need to find the least common multiple (LCM) of the numbers 6 (for the cups) and 8 (for the plates). The LCM will give us the smallest number of items that is a multiple of both 6 and 8.

### Step 1: Prime Factorization
- The prime factorization of 6 is \(2 \times 3\).
- The prime factorization of 8 is \(2^3\).

### Step 2: LCM Calculation
To find the LCM, take the highest power of each prime number that appears in the factorizations:
- The highest power of 2 is \(2^3\).
- The highest power of 3 is \(3\).

Therefore, the LCM is:
\[
\text{LCM} = 2^3 \times 3 = 8 \times 3 = 24
\]

### Step 3: Determine the Number of Packages
- For the cups, since they are sold 6 to a package, you need:
\[
\frac{24}{6} = 4 \text{ packages of cups}
\]
- For the plates, since they are sold 8 to a package, you need:
\[
\frac{24}{8} = 3 \text{ packages of plates}
\]

### Answer
The least number of packages of cups you would need to buy is **4**.

Would you like more details or have any questions? Here are some related questions:

1. What is the general method to find the LCM of two numbers?
2. How do you apply the concept of LCM to different real-world scenarios?
3. What would happen if the number of items in each package was a prime number?
4. How does the LCM differ from the greatest common divisor (GCD)?
5. Can you explain how to find the LCM of three or more numbers?

**Tip:** When trying to match quantities in different packaging sizes, always use the LCM to find the smallest common quantity.

Learn how to calculate the least number of packages of cups needed for a party using the concept of least common multiple (LCM). Follow a step-by-step solution to find the LCM of 6 and 8, ensuring you have an equal number of cups and plates. Suitable for students in Grades 5-7.

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