UGC NET Questions (Paper – 1)

A number divisible by 4 must be divisible by 2, so it is even; no number that is a multiple of 4 can be odd. Likewise, any number divisible by 6 must also be divisible by 2, because 6 = 2 × 3, so divisibility by 6 implies divisibility by 2. Therefore, either statement by itself guarantees that x is even. Since each statement is independently enough to answer the question, each alone is sufficient.

Option A:
Option A underestimates statement (2), but divisibility by 6 already includes divisibility by 2, so statement (2) is also sufficient.

Option B:
Option B reverses the mistake by underestimating statement (1), though any multiple of 4 is clearly even.

Option C:
Option C claims that only the combination of statements can decide parity, which is wrong because each provides its own independent guarantee of evenness.

Option D:
Option D correctly recognises that both conditions individually ensure x is even, making each statement sufficient on its own.

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.

A prime number is an integer greater than 1 with exactly two distinct positive divisors, namely 1 and the number itself. This definition excludes 1 because it has only one positive divisor and excludes composite numbers because they have more than two divisors. In mathematical reasoning, primes are considered the building blocks of the integers. Thus, the description in the question is the standard definition of a prime number.

Option A:
Composite numbers are positive integers greater than 1 that have more than two distinct positive divisors. For example, 6 is composite because it is divisible by 1, 2, 3 and 6. Since the question clearly states "exactly two distinct positive divisors," composite numbers do not fit this requirement. Therefore, this option is incorrect.

Option B:
A multiple is a number that can be obtained by multiplying a given number by an integer, such as 12 being a multiple of 3. This term describes a relationship between numbers and does not specify how many divisors a number has. Hence, it does not answer the definition provided in the stem.

Option C:
Prime is correct because a prime has no positive divisors other than 1 and itself, satisfying the condition of exactly two distinct positive divisors. Examples include 2, 3, 5, 7 and 11. Recognising prime numbers is important in tests involving factorisation, LCM and HCF. Thus, this option directly matches the given description.

Option D:
Even numbers are those divisible by 2, but not all integers greater than 1 divisible by 2 have exactly two divisors. For example, 4 is even but has three positive divisors: 1, 2 and 4. Therefore, "even" is not equivalent to the definition provided and cannot be the correct answer.

Statements A, C and D correctly summarise key properties of HCF and LCM. A gives the fundamental relationship a × b = HCF(a, b) × LCM(a, b) for positive integers. C is true because the LCM is defined as the least common multiple of the numbers. D is correct since when one number divides the other, the smaller is their HCF and the larger is their LCM. B is false because the HCF cannot exceed the smaller number, and E is false because distinct primes have HCF equal to 1, not one of them, so A, C and D only are correct.

Option A:
Option A is incomplete because it omits D, failing to mention the simple special case where one number divides the other, which is frequently used in problems. Even though it includes two true statements, it does not list all of them.

Option B:
Option B is correct as it gathers all the true properties given, including the product relationship, the multiple property of LCM and the special divisor case, while excluding the incorrect descriptions in B and E.

Option C:
Option C is wrong because it includes E, which misstates the HCF of distinct prime numbers. By adding this false statement, it makes the combination invalid.

Option D:
Option D is incomplete since it leaves out A, the central product formula linking HCF and LCM, so it does not provide the full set of correct statements among those listed.

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.

Statements A, C and D all describe standard facts from elementary number theory. Every prime greater than 2 is odd because 2 is the only even prime. Composite numbers must have at least one additional divisor besides 1 and themselves, giving at least three distinct positive divisors. There are infinitely many primes, a result known since Euclid’s proof. Statements B and E are false because 1 is neither prime nor composite and many even numbers greater than 2, such as 4, 6 and 8, are not prime, so A, C and D only are correct.

Option A:
Option A is correct because it picks exactly those statements that are true and excludes the common misconceptions in B and E. It recognises that 1 is excluded from the definition of primes and that even numbers beyond 2 are usually composite. By including C, it also captures a key structural property of composite numbers.

Option B:
Option B is incorrect as it omits D, thereby ignoring the important theorem that there are infinitely many primes. Even though it includes two true statements, it fails to include all the correct ones.

Option C:
Option C is wrong because it brings in E, which incorrectly claims that every even number greater than 2 is prime. Introducing a clearly false statement alongside A and D makes the overall combination incorrect.

Option D:
Option D is incomplete since it includes only C and D and leaves out A. While C and D are true, this option fails to capture the basic fact that primes greater than 2 are odd, so it does not list the full set of correct statements.

The contrapositive of a statement of the form “If p then q” is “If not q then not p,” and these two are logically equivalent. Here, p is “the number is divisible by 6” and q is “the number is divisible by 3.” The contrapositive is therefore “if a number is not divisible by 3, then it is not divisible by 6.” This preserves the logical structure of the original implication while reversing and negating its components.

Option A:
Option A negates p but not q and forms the inverse “If not p then not q,” which is not logically equivalent to the original statement in general.

Option B:
Option B correctly follows the contrapositive pattern ¬q → ¬p, matching the structure of the initial implication about divisibility. It reflects the fact that if a number is not divisible by 3, it cannot possibly be divisible by 6.

Option C:
Option C is the converse q → p, which need not be true simply because the original implication is true; many numbers are divisible by 3 but not by 6.

Option D:
Option D contradicts the original statement directly by combining p with not q, which cannot be true if the original implication holds.

A prime number has exactly two distinct positive divisors, 1 and itself. The numbers 11, 13 and 17 each have no divisors other than 1 and themselves and are therefore prime. The number 15, however, is divisible by 1, 3, 5 and 15, so it has more than two divisors and is composite. Hence 15 is the odd one out in the list.

Option A:
Option A, 11, is a prime number because its only positive divisors are 1 and 11. It shares this property with 13 and 17 and therefore does not stand out in this group.

Option B:
Option B, 13, is also prime and fits the same classification as 11 and 17. It does not have additional factors that would make it different.

Option C:
Option C, 15, clearly deviates from the prime pattern, since it can be expressed as 3 × 5. Its composite nature makes it unique among the options and justifies treating it as the different number.

Option D:
Option D, 17, remains prime since no integer between 2 and 16 divides it without remainder. It aligns with 11 and 13 rather than differing from them.

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.

Statement B is true because adding an odd and an even number always yields an odd result. C is correct since the product of two odd numbers is odd. D is also correct because an odd times an even number always has a factor 2 and so is even. A is false because the sum of two odd integers is even, and E is false because not every even number is divisible by 3. Hence B, C and D only are correct.

Option A:
Option A is incomplete as it leaves out D and therefore does not include the important rule about the product of an odd and even number, which is regularly tested in parity questions.

Option B:
Option B is correct because it gathers all three standard parity rules, covering sums and products of odd and even numbers, while excluding A and E, which misstate these relationships.

Option C:
Option C is incorrect since it includes A, which wrongly claims that two odd numbers add to an odd result, contradicting parity arithmetic.

Option D:
Option D is wrong as it brings in E, which incorrectly generalises divisibility by 3 to all even numbers, and so mixes a false divisibility statement with otherwise correct parity rules.