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The problem asks for the least number of packages of hot dogs and buns needed to have an equal number of hot dogs and buns, given that:

- Hot dogs come in packs of 10.
- Hot dog buns come in packs of 8.

This is essentially a problem of finding the least common multiple (LCM) of 10 and 8, which will give the smallest number that is divisible by both 10 and 8.

### Steps:
1. **Prime factorization of 10**:  
   \( 10 = 2 \times 5 \)
   
2. **Prime factorization of 8**:  
   \( 8 = 2^3 \)
   
3. The LCM takes the highest powers of all prime factors involved:
   - For 2, the highest power is \( 2^3 \).
   - For 5, the highest power is \( 5 \).

4. The LCM is:
   \[
   LCM(10, 8) = 2^3 \times 5 = 8 \times 5 = 40
   \]

Thus, the least number of hot dogs and buns you can have to get an equal amount is 40.

### Number of packages required:
- For **hot dogs**:  
  Since each pack contains 10 hot dogs, you need \( \frac{40}{10} = 4 \) packs.
  
- For **buns**:  
  Since each pack contains 8 buns, you need \( \frac{40}{8} = 5 \) packs.

### Conclusion:
You need **4 packs of hot dogs** and **5 packs of buns** to have an equal number of hot dogs and buns.

Would you like a detailed breakdown or any further clarification?

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Here are 5 related questions to expand your understanding:
1. How do you find the LCM of two numbers using prime factorization?
2. What is the relationship between LCM and GCD?
3. If hot dogs came in packs of 12, how would the solution change?
4. How do you calculate the LCM of three numbers instead of two?
5. Can you always solve problems like this using LCM, and in which scenarios might it not work?

**Tip**: The LCM is useful in situations where you want to synchronize two different repeating events, such as packing items, meeting schedules, or production cycles.

Hot dogs come in packs of 10, and hot dog buns come in packs of 8. What is the least number of packages of each that you can buy to have an equal number of hot dogs and buns?

Learn how to solve the problem of finding the least number of packages to buy hot dogs and buns, where hot dogs come in packs of 10 and buns in packs of 8. Understand the concept of Least Common Multiple (LCM) and prime factorization. Suitable for students in Grades 6-8.

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