LCM of two expressions is `12x^(2)y^(2)z` and their HCF is `4xy`, if one of them is `4x^(2)y`, then the second expression, is

Options

A. `12 x^(2)y^(2)z`
B. `12 xy^(2)z`
C. `12 xyz`
D. `3yz`

Answer

To find the second expression given the LCM, HCF, and one of the expressions, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- LCM = \( 12x^2y^2z \)
- HCF = \( 4xy \)
- One expression (let's call it \( E_1 \)) = \( 4x^2y \)

2. **Use the Relationship Between LCM, HCF, and the Expressions:**
The relationship states that:
\[ \text{LCM} \times \text{HCF} = E_1 \times E_2 \]
where \( E_2 \) is the second expression we need to find.

3. **Substitute the Known Values:**
Substitute the known values into the equation:
\[ 12x^2y^2z \times 4xy = 4x^2y \times E_2 \]

4. **Calculate the Left Side:**
Calculate the left side:
\[ 12 \times 4 = 48 \]
So, we have:
\[ 48x^2y^2z = 4x^2y \times E_2 \]

5. **Rearrange to Solve for \( E_2 \):**
To find \( E_2 \), divide both sides by \( 4x^2y \):
\[ E_2 = \frac{48x^2y^2z}{4x^2y} \]

6. **Simplify the Expression:**
Simplifying the right side:
\[ E_2 = \frac{48}{4} \cdot \frac{x^2}{x^2} \cdot \frac{y^2}{y} \cdot z = 12y \cdot z = 12xyz \]

7. **Final Expression:**
Therefore, the second expression \( E_2 \) is:
\[ E_2 = 12xy^2z \]