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 determine the least number of coloring books Jeremy should buy such that he has the same number of coloring books and boxes of crayons, we need to establish a common multiple of the quantities in which they are sold: 6 (coloring books per pack) and 8 (crayons per pack).
We'll find the least common multiple (LCM) of 6 and 8. First, we can find the prime factorizations:
- The prime factorization of \(6\) is \(2 \times 3\).
- The prime factorization of \(8\) is \(2^3\).
To find the LCM, we take the highest power of each prime present in the factorizations:
- For \(2\), the highest power is \(2^3\) (from 8).
- For \(3\), the highest power is \(3^1\) (from 6).
Now, we compute the LCM:
\[
\text{LCM} = 2^3 \times 3^1 = 8 \times 3 = 24.
\]
The least common multiple of 6 and 8 is 24. This means that:
- To find the number of packs of coloring books Jeremy needs to buy (since each pack has 6 books):
\[
\text{Number of packs of coloring books} = \frac{24}{6} = 4 \quad (\text{since there are 6 coloring books in a pack}),
\]
which gives \(4 \times 6 = 24\) coloring books.
- To find the number of packs of boxes of crayons (since each pack has 8 crayons):
\[
\text{Number of packs of crayons} = \frac{24}{8} = 3 \quad (\text{since there are 8 crayons in a pack}),
\]
which gives \(3 \times 8 = 24\) boxes of crayons.
Thus, the least number of coloring books Jeremy should buy is
\[
\boxed{24}.
\]
We'll find the least common multiple (LCM) of 6 and 8. First, we can find the prime factorizations:
- The prime factorization of \(6\) is \(2 \times 3\).
- The prime factorization of \(8\) is \(2^3\).
To find the LCM, we take the highest power of each prime present in the factorizations:
- For \(2\), the highest power is \(2^3\) (from 8).
- For \(3\), the highest power is \(3^1\) (from 6).
Now, we compute the LCM:
\[
\text{LCM} = 2^3 \times 3^1 = 8 \times 3 = 24.
\]
The least common multiple of 6 and 8 is 24. This means that:
- To find the number of packs of coloring books Jeremy needs to buy (since each pack has 6 books):
\[
\text{Number of packs of coloring books} = \frac{24}{6} = 4 \quad (\text{since there are 6 coloring books in a pack}),
\]
which gives \(4 \times 6 = 24\) coloring books.
- To find the number of packs of boxes of crayons (since each pack has 8 crayons):
\[
\text{Number of packs of crayons} = \frac{24}{8} = 3 \quad (\text{since there are 8 crayons in a pack}),
\]
which gives \(3 \times 8 = 24\) boxes of crayons.
Thus, the least number of coloring books Jeremy should buy is
\[
\boxed{24}.
\]