UGC NET Questions (Paper – 1)

To find the LCM, we factorize the numbers into primes. The number 12 is 2² × 3 and 18 is 2 × 3². The LCM takes the highest power of each prime appearing in either factorization, giving 2² × 3². This product equals 4 × 9 = 36. Hence, 36 is the smallest number that is a multiple of both 12 and 18.

Option A:
Option A, 36, includes both enough factors of 2 and enough factors of 3 to be divisible by the two given numbers. It is also the smallest such number and so satisfies the definition of the LCM.

Option B:
Option B, 24, is divisible by 12 but not by 18 because 24 ÷ 18 is not an integer. It lacks a second factor of 3.

Option C:
Option C, 30, is divisible by neither 12 nor 18 in an exact sense, as it does not contain the required combination of prime powers.

Option D:
Option D, 42, is divisible by 6 and 7, but not by 12 or 18 without leaving a remainder. It is larger than necessary and not a common multiple in this context.

Statements A and B define HCF and LCM correctly, D notes that prime factorisation is a standard method to compute them and E highlights their application to repeated event and synchronisation problems, all of which are true. Statement C is false because in general a × b = HCF(a, b) × LCM(a, b), not just HCF(a, b). Therefore, the combination A, B, D and E only is the set of correct statements.

Option A:
Option A is incomplete as it omits E and thus fails to mention how HCF and LCM play a role in real exam problems involving repeated cycles and meeting times. Without E, the practical context of these concepts is under-emphasised.

Option B:
Option B is correct since it includes all the true statements and rightly excludes C, which confuses the relationship between product, HCF and LCM. It reflects both the definitions and the application of these notions in aptitude questions.

Option C:
Option C is wrong because it contains C, the incorrect equality, even though it also includes some true statements. The presence of C makes the overall combination mathematically inaccurate.

Option D:
Option D is incorrect as it includes C and omits B, thereby endorsing a false formula while ignoring the correct definition of LCM as the smallest common multiple. This mixture cannot be accepted as correct.

Statement A gives the fundamental relationship between HCF, LCM and the product of two positive integers. Statement C correctly defines LCM as the least positive common multiple. Statement D is also true; co-prime numbers have no common positive factor other than 1. Statement B is false because the HCF cannot exceed the numbers themselves, and E is false since if one number divides the other, the HCF is the smaller number, not necessarily 1. Thus, A, C and D only are correct, matching option A.

Option A:
Option A is correct because it combines the key algebraic relation with the proper descriptions of LCM and co-primality, while excluding B and E, which misunderstand bounds on HCF and the divisor case.

Option B:
Option B is incomplete as it includes only A and C and omits D, thereby failing to highlight the link between co-prime numbers and HCF, which is frequently tested.

Option C:
Option C is incorrect because it drops A and includes only C and D, so it loses the important multiplicative connection between HCF and LCM and provides an incomplete set of correct statements.

Option D:
Option D is wrong since it includes B, which wrongly claims that the HCF is at least as large as each number, contradicting basic number theory, and therefore the combination cannot be accepted.