Statement A (Assertion): The HCF of two numbers is 9 and their LCM is 2016. If one of the numbers is 306, then the other is 54. Statement B (Reason): For any positive integers a and b, we have: Product of two numbers = HCF x LCM 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 and 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.
Question
Statement A (Assertion): The HCF of two numbers is 9 and their LCM is 2016. If one of the numbers is 306, then the other is 54. Statement B (Reason): For any positive integers a and b, we have: Product of two numbers = HCF x LCM 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 and 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.
Solution
First, let's check if the assertion (A) is true. The assertion states that the HCF of two numbers is 9 and their LCM is 2016. If one of the numbers is 306, then the other is 54.
We know that the product of two numbers is equal to the product of their HCF and LCM. So, if we multiply the given HCF and LCM, we get 9*2016 = 18144.
Now, if one of the numbers is 306, then the other number would be 18144/306 = 59.33, which is not equal to 54. Therefore, the assertion (A) is false.
Now, let's check if the reason (R) is true. The reason states that for any positive integers a and b, the product of two numbers is equal to the product of their HCF and LCM. This is a well-known property in number theory and is indeed true.
So, the correct answer is (d) Assertion (A) is false but reason (R) is true.
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