Assertion: The HCF of two numbers is 18 and their product is 3072. Then their LCM is [tex]169[/tex].

Reason: For any two positive integers [tex]a[/tex] and [tex]b[/tex], [tex]\(\operatorname{HCF}(a, b) \times \operatorname{LCM}(a, b) = a \times b\)[/tex].

(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).

(c) Assertion (A) is true but reason (R) is false.

(d) Assertion (A) is false but reason (R) is true.



Answer :

To solve the problem, let's analyze the assertion and the reason step-by-step. We need to find whether the given assertion and reason are true and if the reason is the correct explanation for the assertion.

### Given:
- Highest Common Factor (HCF) of two numbers = 18
- Product of the two numbers = 3072
- It's asserted that the Least Common Multiple (LCM) of these two numbers equals to 169

### Reason:
For any two positive integers [tex]\( a \)[/tex] and [tex]\( b \)[/tex],
[tex]\[ \text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b \][/tex]

### Step-by-Step Solution:

1. Calculate the LCM using the given formula:
According to the relationship between HCF and LCM:
[tex]\[ \text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b \][/tex]

2. Substitute the given values into the equation:
Given, [tex]\(\text{HCF} = 18\)[/tex] and the product [tex]\(a \times b = 3072\)[/tex]:
[tex]\[ 18 \times \text{LCM}(a, b) = 3072 \][/tex]

3. Solve for the LCM:
[tex]\[ \text{LCM}(a, b) = \frac{3072}{18} \][/tex]

[tex]\[ \text{LCM}(a, b) = \frac{3072}{18} = 170.6667 \][/tex]

The calculated LCM is approximately 170.67, not 169.

4. Evaluate the Assertion:
The assertion states that the LCM is 169. From our calculations, the LCM does not equal 169. Therefore, the assertion is false.

5. Evaluate the Reason:
The reason provided is:
[tex]\[ \text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b \][/tex]
This is a standard mathematical property and hence is true.

### Conclusion:
- The assertion "The HCF of two numbers is 18 and their product is 3072. Then their LCM = 169." is false.
- The reason "For any two positive integers [tex]\(a\)[/tex] and [tex]\(b\)[/tex] [tex]\(\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b\)[/tex]" is true.

Given these findings, the correct answer is:

(d) Assertion (A) is false but reason (R) is true.