The HCF of 85 and 153 can be expressed in the form of 85m - 153. Calculate the value of m.

Options

A. 1
B. 5
C. `-1`
D. 2

Answer

To solve the problem, we need to find the value of \( m \) such that the highest common factor (HCF) of 85 and 153 can be expressed in the form \( 85m - 153 \).

### Step-by-Step Solution:

1. **Find the HCF of 85 and 153:**
- First, we will find the prime factorization of both numbers.
- For 85:
- \( 85 = 5 \times 17 \)
- For 153:
- \( 153 = 3 \times 51 \)
- \( 51 = 3 \times 17 \)
- Thus, \( 153 = 3 \times 3 \times 17 = 3^2 \times 17 \)
- The common prime factor between 85 and 153 is 17.
- Therefore, the HCF of 85 and 153 is \( 17 \).

2. **Set up the equation using the HCF:**
- According to the problem, we can express the HCF in the form \( 85m - 153 \).
- Therefore, we have:
\[ 17 = 85m - 153 \]

3. **Rearranging the equation:**
- To isolate \( 85m \), we add 153 to both sides:
\[ 85m = 17 + 153 \]
- Simplifying the right side:
\[ 85m = 170 \]

4. **Solving for \( m \):**
- Now, we divide both sides by 85 to find \( m \):
\[ m = \frac{170}{85} \]
- Simplifying the fraction:
\[ m = 2 \]

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