To solve this problem, we need to determine how many packs of wieners and buns are required so that the students sell more than 300 hot dogs without any leftovers.

Given:

• Each pack of wieners contains 12 wieners.
• Each pack of buns contains 8 buns.
• They expect to sell more than 300 hot dogs.

Step 1: Finding the Least Common Multiple (LCM)

Since the number of wieners and buns in each pack is different, we need to ensure that the total number of wieners and buns are equal. To do this, we find the least common multiple (LCM) of 12 (wieners per pack) and 8 (buns per pack).

• Factors of 12: $12 = 2^2 \times 3$
• Factors of 8: $8 = 2^3$

The LCM is found by taking the highest powers of all prime factors:

• LCM of 12 and 8 = $2^3 \times 3 = 24$

This means that every time you buy 2 packs of wieners (24 wieners) and 3 packs of buns (24 buns), you will have exactly 24 hot dogs with no leftovers.

Step 2: Scaling to More Than 300 Hot Dogs

To determine how many packs are needed to sell more than 300 hot dogs, we divide 300 by 24 (the number of hot dogs per set of packs):

$\frac{300}{24} = 12.5$

Since they need to sell more than 300 hot dogs, we round up to 13 sets of 24 hot dogs.

Step 3: Calculating the Number of Packs

Each set consists of:

• 2 packs of wieners (24 wieners)
• 3 packs of buns (24 buns)

For 13 sets:

• Number of wiener packs = $13 \times 2 = 26$ packs
• Number of bun packs = $13 \times 3 = 39$ packs

Final Answer:

The students should buy 26 packs of wieners and 39 packs of buns to sell more than 300 hot dogs without any leftovers.

---

Would you like more details on any part of the explanation?

Here are 5 related questions:

1. How do you find the least common multiple (LCM) of two numbers?
2. What happens if the number of wieners and buns in a pack are the same?
3. How would the solution change if they needed to sell exactly 300 hot dogs?
4. How can you use prime factorization to find the LCM?
5. What are other scenarios where finding the LCM is useful?

Tip: When working with different pack sizes, always look for the LCM to balance the quantities efficiently!