HCF of two numbers – UGC NET Question

The highest common factor (HCF) is the largest number that divides both given numbers exactly. The factors of 24 include 1, 2, 3, 4, 6, 8, 12 and 24, while the factors of 36 include 1, 2, 3, 4, 6, 9, 12, 18 and 36. The common factors are 1, 2, 3, 4, 6 and 12, of which 12 is the largest. Therefore, the HCF of 24 and 36 is 12.

Option A:
4 is a common factor of both numbers, but it is not the largest. Since there exists a greater common factor, namely 12, choosing 4 would underestimate the HCF. Thus, this option is incorrect.

Option B:
6 is also a common factor but not the highest one. Although 6 divides both 24 and 36, 12 is even larger and still divides both exactly. Therefore, 6 cannot represent the highest common factor.

Option C:
8 is a factor of 24 but not of 36, because 36 ÷ 8 is not an integer. Since it does not divide both numbers, it cannot be a common factor at all, let alone the highest. Hence, this option must be rejected.

Option D:
12 is correct because it is the greatest integer that divides both 24 and 36 without remainder. Recognising HCF and LCM is essential in simplifying fractions and solving many number-based aptitude problems. Therefore, this option correctly answers the question.