If a and b are positive integers, then what is (((a)/(HCF(a,b)),(b)/(HCF(a,b)))` equal to -

Options

A. a
B. b
C. 1
D. `(a)/(HCF(a,b))`

Answer

To solve the problem, we need to find the value of \(\left(\frac{a}{\text{HCF}(a,b)}, \frac{b}{\text{HCF}(a,b)}\right)\).

### Step-by-Step Solution:

1. **Understanding HCF**:
The HCF (Highest Common Factor) of two numbers \(a\) and \(b\) is the largest positive integer that divides both \(a\) and \(b\) without leaving a remainder.

2. **Expressing \(a\) and \(b\)**:
Let's denote \(d = \text{HCF}(a, b)\). We can express \(a\) and \(b\) in terms of \(d\):
\[ a = d \cdot m \quad \text{and} \quad b = d \cdot n \]
where \(m\) and \(n\) are coprime integers (i.e., \(\text{HCF}(m, n) = 1\)).

3. **Calculating \(\frac{a}{\text{HCF}(a,b)}\)**:
Substitute \(d\) into the expression:
\[ \frac{a}{\text{HCF}(a,b)} = \frac{d \cdot m}{d} = m \]

4. **Calculating \(\frac{b}{\text{HCF}(a,b)}\)**:
Similarly, for \(b\):
\[ \frac{b}{\text{HCF}(a,b)} = \frac{d \cdot n}{d} = n \]

5. **Finding the HCF of \(m\) and \(n\)**:
Since \(m\) and \(n\) are coprime, we have:
\[ \text{HCF}(m, n) = 1 \]

6. **Final Result**:
Therefore, the expression \(\left(\frac{a}{\text{HCF}(a,b)}, \frac{b}{\text{HCF}(a,b)}\right)\) simplifies to:
\[ (m, n) \quad \text{where} \quad \text{HCF}(m, n) = 1 \]
Hence, the final answer is:
\[ \text{HCF}\left(\frac{a}{\text{HCF}(a,b)}, \frac{b}{\text{HCF}(a,b)}\right) = 1 \]