If the HCF of 408 and 1032 is expressible in the form `1032\ m-408xx5` , find `m` .

Answer

To solve the problem, we need to find the value of \( m \) such that the HCF of 408 and 1032 can be expressed in the form \( 1032m - 408 \times 5 \).

### Step-by-Step Solution:

1. **Find the HCF of 408 and 1032 using the Euclidean Algorithm**:
- Start with the larger number 1032 and divide it by the smaller number 408.
- \( 1032 = 408 \times 2 + 216 \)
- Here, 408 is the divisor, 2 is the quotient, and 216 is the remainder.

2. **Continue the process**:
- Now, take 408 and divide it by the remainder 216.
- \( 408 = 216 \times 1 + 192 \)

3. **Repeat the process**:
- Now take 216 and divide it by 192.
- \( 216 = 192 \times 1 + 24 \)

4. **Continue until the remainder is 0**:
- Now take 192 and divide it by 24.
- \( 192 = 24 \times 8 + 0 \)
- Since the remainder is now 0, the last non-zero remainder is the HCF.

5. **Identify the HCF**:
- The HCF of 408 and 1032 is 24.

6. **Set up the equation**:
- We know that the HCF can be expressed as:
\[ HCF = 1032m - 408 \times 5 \]
- Substitute the HCF value:
\[ 24 = 1032m - 2040 \]

7. **Rearrange the equation**:
- Add 2040 to both sides:
\[ 24 + 2040 = 1032m \]
\[ 2064 = 1032m \]

8. **Solve for \( m \)**:
- Divide both sides by 1032:
\[ m = \frac{2064}{1032} \]
- Simplifying this gives:
\[ m = 2 \]

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