Find th HCF of `16 a ^(4) b ^(3) x ^(3) 24 b ^(2) m ^(3) n ^(4) y and 20 a ^(2) b ^(3) nx ^(3)`

Options

A. `4a ^(2) b ^(2) xy`
B. `4b ^(2)`
C. `8a^(2) b ^(3) n`
D. None of these

Answer

To find the HCF (Highest Common Factor) of the given expressions, we will follow these steps:

### Step 1: Identify the coefficients and variables in each term
The terms we need to find the HCF for are:
1. \( 16 a^4 b^3 x^3 \)
2. \( 24 b^2 m^3 n^4 y \)
3. \( 20 a^2 b^3 n x^3 \)

### Step 2: Find the HCF of the coefficients
The coefficients are:
- 16
- 24
- 20

To find the HCF of these numbers, we can list their factors:
- Factors of 16: \( 1, 2, 4, 8, 16 \)
- Factors of 24: \( 1, 2, 3, 4, 6, 8, 12, 24 \)
- Factors of 20: \( 1, 2, 4, 5, 10, 20 \)

The common factors are \( 1, 2, 4 \). The highest of these is \( 4 \).

### Step 3: Find the HCF of the variables
Now we will find the HCF for each variable:
- For \( a \):
- \( a^4 \) (from the first term)
- \( a^0 \) (from the second term, as \( a \) is not present)
- \( a^2 \) (from the third term)

The HCF is \( a^{\min(4, 0, 2)} = a^0 = 1 \).

- For \( b \):
- \( b^3 \) (from the first term)
- \( b^2 \) (from the second term)
- \( b^3 \) (from the third term)

The HCF is \( b^{\min(3, 2, 3)} = b^2 \).

- For \( x \):
- \( x^3 \) (from the first term)
- \( x^0 \) (from the second term, as \( x \) is not present)
- \( x^3 \) (from the third term)

The HCF is \( x^{\min(3, 0, 3)} = x^0 = 1 \).

- For \( m \):
- \( m^0 \) (from the first term, as \( m \) is not present)
- \( m^3 \) (from the second term)
- \( m^0 \) (from the third term, as \( m \) is not present)

The HCF is \( m^{\min(0, 3, 0)} = m^0 = 1 \).

- For \( n \):
- \( n^0 \) (from the first term, as \( n \) is not present)
- \( n^4 \) (from the second term)
- \( n^1 \) (from the third term)

The HCF is \( n^{\min(0, 4, 1)} = n^0 = 1 \).

- For \( y \):
- \( y^0 \) (from the first term, as \( y \) is not present)
- \( y^1 \) (from the second term, as \( y \) is not present)
- \( y^0 \) (from the third term, as \( y \) is not present)

The HCF is \( y^{\min(0, 0, 0)} = y^0 = 1 \).

### Step 4: Combine the HCFs
Now we combine the HCF of the coefficients and the variables:
- Coefficient HCF: \( 4 \)
- Variable HCF: \( a^0 b^2 x^0 m^0 n^0 y^0 = b^2 \)

Thus, the HCF of the given terms is:
\[ \text{HCF} = 4 b^2 \]

### Final Answer
The HCF of \( 16 a^4 b^3 x^3, 24 b^2 m^3 n^4 y, \) and \( 20 a^2 b^3 n x^3 \) is \( 4 b^2 \).

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