If HCF $(858,325)=858m+325n$, then:

Question

Accepted Answer

To determine why Option 1 is the correct answer, we need to understand the context of the equation given in the question: $ \text{HCF}(858, 325) = 858m + 325n $. This equation is derived from the linear combination of integers, which is a fundamental concept in number theory related to the Euclidean algorithm.

### Explanation for Option 1:

1. **Understanding HCF and Linear Combination:**
   - The Highest Common Factor (HCF) of two numbers is the largest integer that divides both numbers without leaving a remainder.
   - The equation $ 858m + 325n = \text{HCF}(858, 325) $ represents a linear combination of 858 and 325 that equals their HCF.

2. **Calculating the HCF:**
   - Using the Euclidean algorithm, we find the HCF of 858 and 325:
     - $ 858 \div 325 = 2 $ remainder $ 208 $
     - $ 325 \div 208 = 1 $ remainder $ 117 $
     - $ 208 \div 117 = 1 $ remainder $ 91 $
     - $ 117 \div 91 = 1 $ remainder $ 26 $
     - $ 91 \div 26 = 3 $ remainder $ 13 $
     - $ 26 \div 13 = 2 $ remainder $ 0 $
   - The HCF is 13.

3. **Verifying Option 1:**
   - We need to find integers $ m $ and $ n $ such that $ 858m + 325n = 13 $.
   - Using the extended Euclidean algorithm, we backtrack to express 13 as a linear combination:
     - $ 13 = 91 - 3 \times 26 $
     - Substitute back to express 26 in terms of 117 and 91, and so on, until we express 13 in terms of 858 and 325.
   - After performing these calculations, we find that $ m = 11 $ and $ n = -29 $ satisfy the equation.

### Explanation for Other Options:

2. **Option 2: $ m = 29, n = -11 $**
   - Substituting these values into the equation $ 858m + 325n = 13 $ does not satisfy the equation. The linear combination does not equal 13, hence this option is incorrect.

3. **Option 3: $ m = 11, n = 29 $**
   - Substituting these values into the equation $ 858m + 325n = 13 $ results in a different value, not equal to 13. Therefore, this option is incorrect.

4. **Option 4: $ m = -29, n = 11 $**
   - Similarly, substituting these values into the equation does not yield 13. The linear combination does not satisfy the equation, making this option incorrect.

In conclusion, Option 1 is the correct answer because the values $ m = 11 $ and $ n = -29 $ satisfy the equation $ 858m + 325n = 13 $, which is the HCF of 858 and 325. The other options do not satisfy this equation, confirming that Option 1 is indeed correct.