If d is the HCF of a and b, then `d = lambda a + mu b` where :

Options

A. `lambda and mu` are uniquely determined
B. `lambda and mu` are both positive
C. `lambda and mu` are both negative
D. one of the `lambda` and `mu` is negative and the other is positive

Answer

To solve the problem, we need to understand the relationship between the highest common factor (HCF) of two numbers \( a \) and \( b \) and the coefficients \( \lambda \) and \( \mu \) in the equation \( d = \lambda a + \mu b \), where \( d \) is the HCF of \( a \) and \( b \).

### Step-by-Step Solution:

1. **Identify the Values of \( a \) and \( b \)**:
Let's assume \( a = 4 \) and \( b = 2 \).

2. **Calculate the HCF**:
The HCF of \( 4 \) and \( 2 \) is \( 2 \). Thus, \( d = 2 \).

3. **Set Up the Equation**:
According to the problem, we have:
\[ d = \lambda a + \mu b \]
Substituting the known values:
\[ 2 = \lambda(4) + \mu(2) \]
This simplifies to:
\[ 2 = 4\lambda + 2\mu \]

4. **Rearranging the Equation**:
We can divide the entire equation by \( 2 \) to simplify:
\[ 1 = 2\lambda + \mu \]

5. **Assume Values for \( \lambda \)**:
Let's test some values for \( \lambda \):
- **Case 1**: Assume \( \lambda = 1 \):
\[ 1 = 2(1) + \mu \implies \mu = 1 - 2 = -1 \]
- **Case 2**: Assume \( \lambda = 3 \):
\[ 1 = 2(3) + \mu \implies \mu = 1 - 6 = -5 \]

6. **Analyze the Results**:
From the cases we tested:
- In Case 1, \( \lambda = 1 \) (positive) and \( \mu = -1 \) (negative).
- In Case 2, \( \lambda = 3 \) (positive) and \( \mu = -5 \) (negative).

7. **Conclusion**:
In both cases, we find that one coefficient is positive and the other is negative. Therefore, the correct option is that one is negative and the other is positive.

### Final Answer:
The answer is: **One is negative and the other is positive**.