If the 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

Answer

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

### Step-by-Step Solution:

1. **Find the HCF of 65 and 117:**
- First, we will find the prime factorization of both numbers.
- The prime factorization of 65 is \( 5 \times 13 \).
- The prime factorization of 117 is \( 3^2 \times 13 \).
- The common prime factor is 13.
- Therefore, the HCF of 65 and 117 is \( 13 \).

2. **Set up the equation:**
- According to the problem, we need to express the HCF in the form \( 65m - 117 \).
- We can set up the equation:
\[ 65m - 117 = 13 \]

3. **Solve for 'm':**
- Rearranging the equation gives:
\[ 65m = 13 + 117 \]
- Simplifying the right side:
\[ 65m = 130 \]
- Now, divide both sides by 65:
\[ m = \frac{130}{65} \]
- Simplifying gives:
\[ m = 2 \]

4. **Conclusion:**
- The value of \( m \) is \( 2 \).

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

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