Question

For any two positive integers ‘a’ and ‘b’:

• A. $a = \text{HCF}(a, b) \times b$
• B. $a \times b = \text{LCM}(a, b)$
• C. $b = \text{HCF}(a, b) \times \text{LCM}(a, b)$
• D. $a \times b = \text{HCF}(a, b) \times \text{LCM}(a, b)$

Hint:

The product of two numbers is always equal to the product of their HCF and LCM.

Answer 1

The Correct Option is D

Step 1: Recall the relationship between HCF and LCM.
For any two positive integers $a$ and $b$, the product of the two numbers is equal to the product of their HCF (Highest Common Factor) and LCM (Least Common Multiple).
\[ a \times b = \text{HCF}(a, b) \times \text{LCM}(a, b) \]

Step 2: Verification with example.
Let $a = 12$ and $b = 18$.
Then, $\text{HCF}(12, 18) = 6$ and $\text{LCM}(12, 18) = 36$.
Now, check: \[ \text{HCF}(a, b) \times \text{LCM}(a, b) = 6 \times 36 = 216 \] and \[ a \times b = 12 \times 18 = 216 \] Both sides are equal, hence the relation holds true.

Step 3: Conclusion.
Therefore, the correct relation between $a$, $b$, $\text{HCF}$, and $\text{LCM}$ is: \[ a \times b = \text{HCF}(a, b) \times \text{LCM}(a, b) \]