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.

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.

To find the LCM, we factor each number into primes. The factorization of 6 is 2 × 3, of 8 is 2³ and of 9 is 3². The LCM must contain the highest powers of each prime present, so it is 2³ × 3² = 8 × 9 = 72. This is the smallest number divisible by 6, 8 and 9 without remainder.

Option A:
Option A, 36, is divisible by 6 and 9 but not by 8, because 36 ÷ 8 is not an integer. Thus, it fails the requirement of being a common multiple of all three numbers.

Option B:
Option B correctly uses the highest powers of primes appearing in the factorization of each number. By combining 2³ and 3², we ensure divisibility by 6, 8 and 9, and 72 is the least such number.

Option C:
Option C, 48, is divisible by 6 and 8 but not by 9, as 48 ÷ 9 is not an integer. Therefore, it cannot be the LCM.

Option D:
Option D, 54, is divisible by 6 and 9 but not by 8, since 54 ÷ 8 is not a whole number. It is a common multiple of only two of the three numbers.