To find out how many hot dogs Arpitha can serve at her party with the given constraints, we need to determine the least common multiple (LCM) of the number of hot dogs in a package and the number of buns in a package.
- Hot dogs come in packages of 8.
- Hot dog buns come in packages of 12.
We are looking for the LCM of 8 and 12.
Step 1: Determine the prime factorization of both numbers.
- The prime factorization of 8 is \(2^3\).
- The prime factorization of 12 is \(2^2 \times 3^1\).
Step 2: Use these factorizations to find the LCM.
To find the LCM, take the highest power of each prime that appears in the factorizations:
- For \(2\): The higher power is \(2^3\).
- For \(3\): The higher power is \(3^1\).
Now multiply these: \[ \text{LCM} = 2^3 \times 3^1 = 8 \times 3 = 24 \]
Step 3: Calculate the minimum number of packages required.
Now that we know the LCM is 24, we can determine how many packages of hot dogs and buns Arpitha needs.
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For hot dogs: \[ \text{Number of hot dog packages} = \frac{24 \text{ hot dogs}}{8 \text{ hot dogs/package}} = 3 \text{ packages} \]
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For buns: \[ \text{Number of buns packages} = \frac{24 \text{ buns}}{12 \text{ buns/package}} = 2 \text{ packages} \]
Conclusion:
Arpitha can serve 24 hot dogs at her party.
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