HCF of Two Numbers – UGC NET Question

The highest common factor is the largest number that divides both given numbers exactly. The prime factorization of 18 is 2 × 3 × 3 and of 30 is 2 × 3 × 5. The common prime factors are 2 and 3, and multiplying them gives 2 × 3 = 6. Since no larger number divides both 18 and 30 without remainder, 6 is the highest common factor.

Option A:
Option A correctly uses the idea of common prime factors and combines 2 and 3 to produce 6. It respects both the definition of a factor and the requirement that the factor be as large as possible while still dividing both numbers exactly. This makes 6 the appropriate highest common factor for 18 and 30.

Option B:
Option B, 3, is a common factor but not the greatest one because 6 also divides both numbers and is larger than 3. Choosing 3 ignores the possibility of combining more common prime factors.

Option C:
Option C, 5, divides 30 but does not divide 18, so it fails the basic requirement that the highest common factor must be a factor of both numbers. It is therefore not a common factor at all in this pair.

Option D:
Option D, 9, divides 18 but not 30 because 30 ÷ 9 does not yield an integer. Hence, it also cannot be regarded as a common factor of the two numbers.