To solve this problem, we need to determine the least common multiple (LCM) of 8 (number of hot dogs in a pack) and 12 (number of buns in a pack). This will give us the smallest number of hot dogs and buns that Milan could have bought, ensuring that he bought an equal number of hot dogs and buns.

Step 1: Find the LCM of 8 and 12

• The prime factorization of 8 is $8 = 2^3$.
• The prime factorization of 12 is $12 = 2^2 \times 3$.

The LCM is found by taking the highest powers of all prime factors: $LCM(8, 12) = 2^3 \times 3 = 24$ So, the smallest number of hot dogs and buns Milan could have bought is 24.

Step 2: Determine how many packs he bought

• For hot dogs: Each pack contains 8 hot dogs, so Milan bought $\frac{24}{8} = 3$ packs of hot dogs.
• For buns: Each pack contains 12 buns, so Milan bought $\frac{24}{12} = 2$ packs of buns.

Step 3: Summary

• Packs of hot dogs Milan bought: 3 packs
• Packs of buns Milan bought: 2 packs
• Number of hot dogs Milan bought: 24 hot dogs

Let me know if you'd like more details or have any questions. Here are some related questions to explore:

1. What is the LCM of 8 and 16, and how does it change the number of packs?
2. If Milan bought 4 packs of hot dogs, how many buns would he need to match the quantity?
3. How would this problem change if buns came in packs of 10 instead of 12?
4. What is the greatest common divisor (GCD) of 8 and 12, and how does it relate to the LCM?
5. If Milan needed at least 50 hot dogs, how many packs of each would he need?

Tip: When finding the least common multiple, always use the highest powers of prime factors from each number.