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.