If `x = 3^2 xx 2` and `y = 2^3 xx 3` and `z = 3^2 xx 4`, then find the LCM of xy, yz and zx.

Options

A. 2592
B. 2416
C. 2748
D. 2936

Answer

To solve the problem step by step, let's first define the values of \( x \), \( y \), and \( z \) as given in the question:

1. **Define the values**:
- \( x = 3^2 \times 2 \)
- \( y = 2^3 \times 3 \)
- \( z = 3^2 \times 4 \)

2. **Calculate \( xy \), \( yz \), and \( zx \)**:
- **Calculate \( xy \)**:
\[ xy = (3^2 \times 2) \times (2^3 \times 3) = 3^2 \times 2^1 \times 2^3 \times 3^1 = 3^{2+1} \times 2^{1+3} = 3^3 \times 2^4 \]
- **Calculate \( yz \)**:
\[ yz = (2^3 \times 3) \times (3^2 \times 4) = 2^3 \times 3^1 \times 3^2 \times 4 = 2^3 \times 3^{1+2} \times 4 = 2^3 \times 3^3 \times 2^2 = 2^{3+2} \times 3^3 = 2^5 \times 3^3 \]
- **Calculate \( zx \)**:
\[ zx = (3^2 \times 4) \times (3^2 \times 2) = 3^2 \times 4 \times 3^2 \times 2 = 3^{2+2} \times 4 \times 2 = 3^4 \times 4 \times 2 = 3^4 \times 2^1 \times 2^2 = 3^4 \times 2^{1+2} = 3^4 \times 2^3 \]

3. **Summarize the results**:
- \( xy = 3^3 \times 2^4 \)
- \( yz = 3^3 \times 2^5 \)
- \( zx = 3^4 \times 2^3 \)

4. **Find the LCM of \( xy \), \( yz \), and \( zx \)**:
- To find the LCM, we take the highest power of each prime factor:
- For \( 2 \): The maximum power is \( 5 \) (from \( yz \)).
- For \( 3 \): The maximum power is \( 4 \) (from \( zx \)).
- Therefore, the LCM is:
\[ \text{LCM} = 2^5 \times 3^4 \]

5. **Calculate the LCM**:
- Calculate \( 2^5 \):
\[ 2^5 = 32 \]
- Calculate \( 3^4 \):
\[ 3^4 = 81 \]
- Now multiply these results:
\[ \text{LCM} = 32 \times 81 \]
- Performing the multiplication:
\[ 32 \times 81 = 2592 \]

Thus, the final answer is:
\[ \text{LCM}(xy, yz, zx) = 2592 \]