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.
### 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.
