If a and b are positive integers, then what is the value of `((alpha)/(HCF(a,b)) ,(b)/(HCF(a,b)))`?

Options

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

Answer

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

### 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 \(b\).

2. **Expressing \(a\) and \(b\)**:
Let \(H = \text{HCF}(a, b)\). We can express \(a\) and \(b\) in terms of \(H\):
\[ a = H \cdot m \quad \text{and} \quad b = H \cdot n \]
where \(m\) and \(n\) are integers that have no common factors (i.e., \(\text{HCF}(m, n) = 1\)).

3. **Substituting into the Expression**:
Now, substitute \(a\) and \(b\) into the expression:
\[ \left(\frac{a}{H}, \frac{b}{H}\right) = \left(\frac{H \cdot m}{H}, \frac{H \cdot n}{H}\right) \]
This simplifies to:
\[ (m, n) \]

4. **Finding the HCF of \(m\) and \(n\)**:
Since \(m\) and \(n\) are coprime (i.e., \(\text{HCF}(m, n) = 1\)), we can conclude that:
\[ \text{HCF}(m, n) = 1 \]

5. **Final Result**:
Therefore, the value of the expression \(\left(\frac{a}{\text{HCF}(a,b)}, \frac{b}{\text{HCF}(a,b)}\right)\) is:
\[ (m, n) \quad \text{where } \text{HCF}(m, n) = 1 \]
This means that the values of \(m\) and \(n\) are such that their HCF is 1, leading us to conclude that the final answer is:
\[ (1, 1) \]

### Conclusion:
The value of \(\left(\frac{a}{\text{HCF}(a,b)}, \frac{b}{\text{HCF}(a,b)}\right)\) is always \((1, 1)\) when \(a\) and \(b\) are expressed in terms of their HCF.