Question
Coloring books are sold in packs of 6, and boxes of crayons are sold in packs of 8.
Jeremy wants to have the same number of coloring books and boxes of crayons, without any leftovers. What is the least number of books he should buy?
Jeremy wants to have the same number of coloring books and boxes of crayons, without any leftovers. What is the least number of books he should buy?
Answers
GPT-4o mini
To find the least number of coloring books Jeremy should buy such that he has an equal number of coloring books and boxes of crayons, we need to determine the least common multiple (LCM) of the pack sizes of each item.
The number of coloring books is in packs of 6, and the number of boxes of crayons is in packs of 8. We need to find the LCM of these two numbers: 6 and 8.
**Step 1: Find the prime factorization of each number.**
- The prime factorization of 6 is:
\[
6 = 2^1 \times 3^1
\]
- The prime factorization of 8 is:
\[
8 = 2^3
\]
**Step 2: Determine the LCM.**
To find the LCM, we take the highest power of each prime that appears in either factorization.
- For the prime number 2:
- Highest power in 6: \(2^1\)
- Highest power in 8: \(2^3\)
- Therefore, we take \(2^3\).
- For the prime number 3:
- Highest power in 6: \(3^1\)
- Highest power in 8: \(3^0\) (which is considered as \(1\))
- Therefore, we take \(3^1\).
Now we can calculate the LCM:
\[
\text{LCM}(6, 8) = 2^3 \times 3^1 = 8 \times 3 = 24.
\]
**Step 3: Calculate the number of packs needed.**
Now that we know the LCM is 24, we need to find out how many packs of each item Jeremy should buy:
- For coloring books:
\[
\text{Number of packs of coloring books} = \frac{24}{6} = 4 \text{ packs}.
\]
- For boxes of crayons:
\[
\text{Number of packs of crayons} = \frac{24}{8} = 3 \text{ packs}.
\]
**Step 4: Calculate the total number of items.**
- Total number of coloring books: \(4 \text{ packs} \times 6 \text{ books/pack} = 24 \text{ coloring books}\).
- Total number of boxes of crayons: \(3 \text{ packs} \times 8 \text{ crayons/pack} = 24 \text{ boxes of crayons}\).
Thus, the least number of coloring books Jeremy should buy is
\[
\boxed{24}.
\]
The number of coloring books is in packs of 6, and the number of boxes of crayons is in packs of 8. We need to find the LCM of these two numbers: 6 and 8.
**Step 1: Find the prime factorization of each number.**
- The prime factorization of 6 is:
\[
6 = 2^1 \times 3^1
\]
- The prime factorization of 8 is:
\[
8 = 2^3
\]
**Step 2: Determine the LCM.**
To find the LCM, we take the highest power of each prime that appears in either factorization.
- For the prime number 2:
- Highest power in 6: \(2^1\)
- Highest power in 8: \(2^3\)
- Therefore, we take \(2^3\).
- For the prime number 3:
- Highest power in 6: \(3^1\)
- Highest power in 8: \(3^0\) (which is considered as \(1\))
- Therefore, we take \(3^1\).
Now we can calculate the LCM:
\[
\text{LCM}(6, 8) = 2^3 \times 3^1 = 8 \times 3 = 24.
\]
**Step 3: Calculate the number of packs needed.**
Now that we know the LCM is 24, we need to find out how many packs of each item Jeremy should buy:
- For coloring books:
\[
\text{Number of packs of coloring books} = \frac{24}{6} = 4 \text{ packs}.
\]
- For boxes of crayons:
\[
\text{Number of packs of crayons} = \frac{24}{8} = 3 \text{ packs}.
\]
**Step 4: Calculate the total number of items.**
- Total number of coloring books: \(4 \text{ packs} \times 6 \text{ books/pack} = 24 \text{ coloring books}\).
- Total number of boxes of crayons: \(3 \text{ packs} \times 8 \text{ crayons/pack} = 24 \text{ boxes of crayons}\).
Thus, the least number of coloring books Jeremy should buy is
\[
\boxed{24}.
\]