Assertion (A): If two numbers are co-primes, then both the numbers may be composite.
Reason (R): Two natural numbers which do not have a common prime factor are called co-primes.
Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
Assertion (A) is true but Reason (R) is false.
Assertion (A) is false but Reason (R) is true.
Answer
Two composite numbers can be co-prime if they do not share any common prime factors. For example, 8 and 9 are both composite, but they are co-prime. So, the Assertion (A) is true.
The Reason (R) is true and it provides the definitive rule that allows for the Assertion to occur. As, the only requirement is the absence of a shared prime factor, the numbers themselves are not required to be prime.
Hence, Both A and R are true and R is the correct explanation of A.
Assertion (A): If a number is divisible by both 2 and 3, then it is divisible by 12.
Reason (R): All the numbers divisible by 6 are even numbers.
Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
Assertion (A) is true but Reason (R) is false.
Assertion (A) is false but Reason (R) is true.
Answer
If a number is divisible by 2 and 3, it is divisible by their least common multiple, which is 6. It is not necessarily divisible by 12. A counterexample is 6 (divisible by 2 and 3, but not 12) or 18. So the Assertion (A) is false.
All the numbers divisible by 6 are multiples of 6. Since 6 is an even number, all its integer multiples are also even. So the Reason (R) is true.
Hence, A is false but R is true.
Assertion (A): L.C.M. of two or more numbers may be one of the numbers.
Reason (R): For two given numbers, product of the numbers = product of their H.C.F. and L.C.M.
Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
Assertion (A) is true but Reason (R) is false.
Assertion (A) is false but Reason (R) is true.
Answer
If one number is a multiple of the other, the larger number is the L.C.M. For example, the L.C.M. of 4 and 12 is 12. So the Assertion (A) is true.
For two given numbers, product of the numbers = product of their H.C.F. and L.C.M. So the Reason (R) is a valid mathematical property.
Both statements are true, but the Reason (R) does not explain the Assertion. The Assertion is true due to the concept of multiples and factors, not because of the product relationship between the H.C.F. and L.C.M.
Hence, Both A and R are true but R is not the correct explanation of A.