If HCF of `65` and `117` is expressible in the form `65m-117`, then the value of `m` is

Options

A. `4`
B. `2`
C. `1`
D. `3`

Explanation

To solve the problem, we need to find the value of \( m \) such that the HCF of \( 65 \) and \( 117 \) can be expressed in the form \( 65m - 117 \).

### Step-by-Step Solution:

1. **Find the HCF of 65 and 117:**
- The prime factorization of \( 65 \) is \( 5 \times 13 \).
- The prime factorization of \( 117 \) is \( 3 \times 39 \) or \( 3 \times 3 \times 13 \).
- The common factor between \( 65 \) and \( 117 \) is \( 13 \).
- Therefore, \( \text{HCF}(65, 117) = 13 \).

2. **Set up the equation:**
- According to the problem, we have \( \text{HCF} = 65m - 117 \).
- We can substitute the HCF we found into the equation:
\[ 13 = 65m - 117 \]

3. **Rearrange the equation to solve for \( m \):**
- Add \( 117 \) to both sides:
\[ 13 + 117 = 65m \]
\[ 130 = 65m \]

4. **Solve for \( m \):**
- Divide both sides by \( 65 \):
\[ m = \frac{130}{65} \]
- Simplifying this gives:
\[ m = 2 \]

### Final Answer:
The value of \( m \) is \( 2 \).