Question
Mr. Thompson is organizing a school field trip and needs to plan for snacks. He decides to pack granola bars and small packs of dried fruit for each student. He has found a vendor that sells granola bars in packs of 8 and dried fruit packs in packs of 12. Mr. Thompson does not want to have any leftover individual snacks. First, Mr. Thompson needs to calculate the least common multiple (LCM) of 8 and 12. Next, Mr. Thompson needs to determine how many packages of each item he should buy to ensure that each student gets the same number of granola bars and dried fruit packs while purchasing the fewest number of packages possible.
Answers
GPT-4o mini
To find the least common multiple (LCM) of 8 and 12, we first find the prime factorization of each number:
- The prime factorization of 8 is \(2^3\).
- The prime factorization of 12 is \(2^2 \times 3^1\).
To calculate the LCM, we take the highest power of each prime number present in the factorizations:
- For the prime 2: the highest power is \(2^3\) (from 8).
- For the prime 3: the highest power is \(3^1\) (from 12).
Now we find the LCM by multiplying these highest powers together:
\[
\text{LCM} = 2^3 \times 3^1 = 8 \times 3 = 24
\]
Thus, the LCM of 8 and 12 is \(24\).
Next, we need to determine how many packages Mr. Thompson should buy for each item.
1. **Granola Bars**:
- Since the granola bars are sold in packs of 8, we need to find how many packs of granola bars are needed to have 24 granola bars:
\[
\frac{24 \text{ (total snacks needed)}}{8 \text{ (per pack)}} = 3 \text{ packs of granola bars}
\]
2. **Dried Fruit Packs**:
- Since the dried fruit packs are sold in packs of 12, we need to find how many packs of dried fruit are needed to have 24 packs of dried fruit:
\[
\frac{24 \text{ (total snacks needed)}}{12 \text{ (per pack)}} = 2 \text{ packs of dried fruit}
\]
**Summary**:
- Mr. Thompson should buy **3 packs of granola bars** and **2 packs of dried fruit** to ensure that each student gets the same number of snacks without any leftovers.
- The prime factorization of 8 is \(2^3\).
- The prime factorization of 12 is \(2^2 \times 3^1\).
To calculate the LCM, we take the highest power of each prime number present in the factorizations:
- For the prime 2: the highest power is \(2^3\) (from 8).
- For the prime 3: the highest power is \(3^1\) (from 12).
Now we find the LCM by multiplying these highest powers together:
\[
\text{LCM} = 2^3 \times 3^1 = 8 \times 3 = 24
\]
Thus, the LCM of 8 and 12 is \(24\).
Next, we need to determine how many packages Mr. Thompson should buy for each item.
1. **Granola Bars**:
- Since the granola bars are sold in packs of 8, we need to find how many packs of granola bars are needed to have 24 granola bars:
\[
\frac{24 \text{ (total snacks needed)}}{8 \text{ (per pack)}} = 3 \text{ packs of granola bars}
\]
2. **Dried Fruit Packs**:
- Since the dried fruit packs are sold in packs of 12, we need to find how many packs of dried fruit are needed to have 24 packs of dried fruit:
\[
\frac{24 \text{ (total snacks needed)}}{12 \text{ (per pack)}} = 2 \text{ packs of dried fruit}
\]
**Summary**:
- Mr. Thompson should buy **3 packs of granola bars** and **2 packs of dried fruit** to ensure that each student gets the same number of snacks without any leftovers.