Question
arranging boys into 12, 15, 18 equal rows and a solid square
Answers
Answered by
GPT-4o mini
To determine how many boys can be arranged into equal rows of 12, 15, and 18, as well as in a solid square formation, we need to find the least common multiple (LCM) of the numbers 12, 15, and 18.
1. **Find the prime factorization of each number:**
- 12: \(2^2 \times 3^1\)
- 15: \(3^1 \times 5^1\)
- 18: \(2^1 \times 3^2\)
2. **Take the highest power of each prime that appears in any of the factorizations:**
- The highest power of \(2\): \(2^2\) (from 12)
- The highest power of \(3\): \(3^2\) (from 18)
- The highest power of \(5\): \(5^1\) (from 15)
3. **Calculate the LCM:**
\[
\text{LCM} = 2^2 \times 3^2 \times 5^1 = 4 \times 9 \times 5
\]
\[
4 \times 9 = 36
\]
\[
36 \times 5 = 180
\]
Thus, the least common multiple of 12, 15, and 18 is 180.
4. **Solid Square Formation:**
A solid square arrangement means the number of boys must be a perfect square. The closest perfect square to 180 is \(13^2 = 169\) and \(14^2 = 196\). The next perfect square after 180 is 196, which isn't a common arrangement with the numbers you provided.
Therefore, the number of boys that can be arranged into rows of 12, 15, 18, and can also be in a solid square layout would be at least 180, but they could not be arranged into a perfect square with just these three given row functions. The viable arrangements based on the conditions given focus only on rows of 12, 15, and 18. If a solid square layout is also desired, the next possible number is 196 after checking against the perfect square criteria.
1. **Find the prime factorization of each number:**
- 12: \(2^2 \times 3^1\)
- 15: \(3^1 \times 5^1\)
- 18: \(2^1 \times 3^2\)
2. **Take the highest power of each prime that appears in any of the factorizations:**
- The highest power of \(2\): \(2^2\) (from 12)
- The highest power of \(3\): \(3^2\) (from 18)
- The highest power of \(5\): \(5^1\) (from 15)
3. **Calculate the LCM:**
\[
\text{LCM} = 2^2 \times 3^2 \times 5^1 = 4 \times 9 \times 5
\]
\[
4 \times 9 = 36
\]
\[
36 \times 5 = 180
\]
Thus, the least common multiple of 12, 15, and 18 is 180.
4. **Solid Square Formation:**
A solid square arrangement means the number of boys must be a perfect square. The closest perfect square to 180 is \(13^2 = 169\) and \(14^2 = 196\). The next perfect square after 180 is 196, which isn't a common arrangement with the numbers you provided.
Therefore, the number of boys that can be arranged into rows of 12, 15, 18, and can also be in a solid square layout would be at least 180, but they could not be arranged into a perfect square with just these three given row functions. The viable arrangements based on the conditions given focus only on rows of 12, 15, and 18. If a solid square layout is also desired, the next possible number is 196 after checking against the perfect square criteria.