UGC NET Questions (Paper – 1)

Statements A, B, D and E are correct standard number-theoretic facts: even integers are divisible by 2, primes greater than 2 are odd, divisibility by 6 implies divisibility by both 2 and 3 and divisibility tests accelerate factor checking in exams. Statement C is false because a composite number has more than two positive divisors, whereas having exactly two is the defining feature of a prime. Therefore, the combination containing A, B, D and E only is correct.

Option A:
Option A is incomplete because it omits E, thereby ignoring the important exam-oriented point that divisibility rules are employed to solve questions quickly. Without E, the applied angle of the topic is underrepresented.

Option B:
Option B is correct as it includes all true statements while leaving out C, which reverses the definition of composite and prime numbers. It aligns with the basic arithmetic knowledge expected at UGC NET level.

Option C:
Option C is wrong because it includes C, the incorrect statement about composites having exactly two divisors, and omits A, a correct definition of even numbers. This combination thus mixes error with omission.

Option D:
Option D is incorrect as it accepts C along with true statements and omits B, failing to state that primes greater than 2 are odd. The presence of C again makes the option unacceptable.

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.

This option is correct because the product of an even integer with any integer is always even. An even number can be written as 2k, and multiplying 2k by any integer m gives 2(km), which is still divisible by 2. Therefore, if at least one factor is even, the product is even.

Option A:
Even numbers are multiples of 2, and when such a number multiplies any integer, the factor of 2 remains in the result. This ensures the product is also a multiple of 2 and hence even. Thus, this option correctly states the property described.

Option B:
An odd product occurs only when both factors are odd, because then neither factor contributes a factor of 2. Here, the condition explicitly says at least one integer is even, so the product cannot be odd. Therefore, this option is wrong.

Option C:
A prime number has exactly two positive divisors, 1 and itself. The product of two integers with one even factor can be composite and need not be prime. So, prime is not guaranteed and is not the correct description.

Option D:
A negative product depends on the signs of the integers, not on whether they are even or odd. An even factor does not force the product to be negative. Hence, negativity is unrelated to the parity condition in the question.

An even number is defined as an integer that can be written in the form 2k, where k is an integer. This means it is exactly divisible by 2 without leaving any remainder. Numbers like −4, 0, 2 and 10 are all even because they satisfy this property. Therefore, the description in the question perfectly matches the standard definition of an even number.

Option A:
Prime numbers are integers greater than 1 that have exactly two distinct positive divisors, 1 and the number itself. While some primes like 2 are even, most primes are not characterised simply by divisibility by 2. The defining property in the question is divisibility by 2 for all such numbers, which corresponds to evenness, not primality. Hence, this option is not correct.

Option B:
Odd numbers are integers that leave a remainder of 1 when divided by 2. They can be expressed as 2k+1 for some integer k. This is the opposite of being exactly divisible by 2, so odd numbers do not fit the description given in the question. Therefore, this option must be rejected.

Option C:
Even numbers are correct because they are precisely those integers that divide by 2 with no remainder. In many aptitude questions, recognising even and odd numbers quickly helps in solving problems on divisibility and parity. Knowing this definition is foundational for reasoning about number patterns and algebraic expressions. Thus, this option accurately answers the question.

Option D:
Composite numbers have more than two positive divisors, such as 4, 6 or 12, but the question is not about the count of divisors. Some composite numbers are even and some are odd, so "composite" is not equivalent to "divisible by 2." As the stem focuses solely on divisibility by 2, composite is not the correct term.

Let the three consecutive even numbers be n − 2, n and n + 2, where n is the middle one. Their sum is (n − 2) + n + (n + 2) = 3n. According to the given condition, 3n = 66, so n = 66 ÷ 3 = 22. Hence, the middle even number is 22.

Option A:
Option A, 20, would give the three numbers 18, 20 and 22, whose sum is 60, not 66. Therefore, it does not satisfy the condition set in the problem.

Option B:
Option B fits perfectly: with 22 as the middle number, the three numbers become 20, 22 and 24, and their sum is 20 + 22 + 24 = 66, matching the given total.

Option C:
Option C, 24, would generate the sequence 22, 24 and 26, whose sum is 72, exceeding the required sum.

Option D:
Option D, 26, would produce 24, 26 and 28, summing to 78, which is also inconsistent with the problem statement.

This option is correct because an even number is defined as any integer that can be written as 2k for some integer k. Being a multiple of 2 means the number fits this form. Therefore, any multiple of 2 is an even number.

Option A:
Even numbers have 2 as a factor and are exactly divisible by 2. Writing the number as 2k shows it is a multiple of 2, which matches the condition in the question. Thus, this option is correct.

Option B:
Odd numbers are not multiples of 2 and leave a remainder of 1 when divided by 2. Since the question specifies multiples of 2, odd numbers cannot be the correct classification.

Option C:
Prime numbers are defined by having exactly two distinct positive divisors. Some primes like 2 are even, but not all multiples of 2 are prime. Many even numbers have more than two divisors, so prime is not the right general term here.

Option D:
Composite numbers have more than two positive divisors and can be even or odd. The phrase "multiple of 2" points specifically to evenness, not to the broader idea of compositeness. Hence, this option is not precise enough.

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 smallest number to be added, we first determine the remainder when 625 is divided by 7. Since 7 × 89 = 623, the remainder is 625 − 623 = 2. A number is divisible by 7 when the remainder becomes zero, so we must add a number that brings the remainder 2 up to 7. Therefore, we add 5 (because 2 + 5 = 7), making 625 + 5 = 630, which is divisible by 7.

Option A:
Option A correctly uses the idea that the required adjustment is 7 minus the remainder. With a remainder of 2, adding 5 eliminates the remainder and yields a number (630) that lies exactly on the 7-times table.

Option B:
Option B, 2, would produce 627, and 627 ÷ 7 still leaves a remainder because 7 × 89 = 623 and 7 × 90 = 630. Thus, 627 is not divisible by 7.

Option C:
Option C, 4, gives 629, which also falls between multiples of 7 and leaves a nonzero remainder, so it does not satisfy the condition.

Option D:
Option D, 6, would lead to 631, which again does not align with a multiple of 7, because the nearest multiples are 630 and 637.

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.

Statements A, B, C and E correctly describe the logic behind classification questions. Such items ask for the outlier, use a clear basis such as number properties or meaning, rely on a single consistent rule for the odd item and are best solved by first identifying what the majority have in common. Statement D is false because well-designed questions typically have exactly one odd item, not two, so that there is a unique correct answer. Hence, the combination that includes A, B, C and E but excludes D is the correct one.

Option A:
Option A is correct since it contains all of the true statements and omits D, which introduces ambiguity by suggesting two odd items. This option best reflects how classification questions are set in UGC NET and similar exams.

Option B:
Option B is incomplete because it omits E, failing to highlight the crucial strategy of first spotting the common feature shared by most items. Without E, the guidance on how to approach such questions is not fully conveyed.

Option C:
Option C is incorrect as it leaves out A and therefore does not explicitly mention that the task is to identify the non-member of the group, which is the central idea of classification items. Even though B, C and E are true, the set is not complete.

Option D:
Option D is wrong because it includes D, the false claim about having two odd items, and omits B, which correctly indicates different possible bases of grouping. By combining a wrong statement with omissions, this option cannot be accepted.

This option is correct because an odd number is defined as an integer that is not divisible by 2. When an odd number is divided by 2, it always leaves a remainder of 1. This matches exactly the condition given in the question. Therefore, an integer not divisible by 2 is called an odd number.

Option A:
An odd number has the form 2k+1 for some integer k and is not divisible by 2. Dividing any odd number by 2 leaves a remainder of 1, which fits the description in the stem. Hence, this option correctly names the type of number described.

Option B:
An even number is divisible by 2 and leaves no remainder on division by 2. The question specifies numbers that are not divisible by 2, so this property does not match. Therefore, this option is not correct here.

Option C:
A prime number is defined by having exactly two distinct positive divisors, 1 and the number itself. Some primes are odd but the concept of primality does not directly refer to divisibility by 2 alone. Hence, prime is not the specific term required by the question.

Option D:
A composite number has more than two positive divisors. It can be even or odd, and its definition is unrelated to divisibility by 2 alone. Therefore, composite is not the best term for numbers that are simply not divisible by 2.

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.

For two positive integers a and b, the product a × b equals HCF(a, b) × LCM(a, b). Here, one number is 24, the HCF is 6 and the LCM is 72. Let the other number be n. Then 24 × n = 6 × 72. The right-hand side is 432, so n = 432 ÷ 24 = 18. Hence, the other number must be 18 to maintain both the given HCF and LCM.

Option A:
Option A, 12, would give 24 × 12 = 288, and since 6 × 72 = 432, the product condition would fail. It also changes the LCM and HCF relationship.

Option B:
Option B satisfies 24 × 18 = 432, matching 6 × 72 exactly. It ensures the pair of numbers has the stated HCF and LCM by the standard product relation.

Option C:
Option C, 36, gives a product of 864, which would require a different LCM or HCF than those given.

Option D:
Option D, 48, yields an even larger product and is inconsistent with the specified values of 6 and 72.