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Question (1)

QN0036472

Which statement correctly describes the Fundamental Theorem of Arithmetic?

(A) Every natural number is prime
(B) Every composite number has a unique factorisation into primes, apart from order
(C) Every integer has exactly two prime factors
(D) Every prime number can be expressed as a product of composite numbers

Correct Answer: B

Explanation:

Every composite number can be expressed as a product of primes, and this factorisation is unique except for the order of the prime factors.

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Question (2)

QN0036473

Which of the following is the prime factorisation of 540?

(A) 2² × 3³ × 5
(B) 2³ × 3² × 5
(C) 2² × 3² × 5²
(D) 2 × 3³ × 5²

Correct Answer: A

Explanation:

540 = 54 × 10 = (2 × 3³) × (2 × 5) = 2² × 3³ × 5.

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Question (3)

QN0036474

If p is a prime number and p divides a², then which conclusion is necessarily true?

(A) p divides a
(B) a divides p
(C) p = a²
(D) a must be prime

Correct Answer: A

Explanation:

If a prime p divides a², then p must divide a.

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Question (4)

QN0036475

The prime factorisation of 840 is:

(A) 2² × 3 × 5 × 7
(B) 2³ × 3 × 5 × 7
(C) 2³ × 3² × 5 × 7
(D) 2² × 3² × 5 × 7

Correct Answer: B

Explanation:

840 = 8 × 105 = 2³ × 3 × 5 × 7.

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Question (5)

QN0036476

Which of the following is irrational?

(A) 7/11
(B) 0.125
(C) √7
(D) 22/7

Correct Answer: C

Explanation:

7 is prime, and the square root of a prime number is irrational. Therefore, √7 is irrational.

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Question (6)

QN0036477

Find the HCF of 84 and 126.

(A) 21
(B) 28
(C) 42
(D) 14

Correct Answer: C

Explanation:

84 = 2² × 3 × 7 and 126 = 2 × 3² × 7. Therefore, HCF = 2 × 3 × 7 = 42.

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Question (7)

QN0036478

Find the LCM of 18 and 45.

(A) 90
(B) 135
(C) 180
(D) 270

Correct Answer: A

Explanation:

18 = 2 × 3² and 45 = 3² × 5. Therefore, LCM = 2 × 3² × 5 = 90.

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Question (8)

QN0036479

If the HCF of two positive integers is 12 and their product is 4320, their LCM is:

(A) 240
(B) 300
(C) 360
(D) 420

Correct Answer: C

Explanation:

HCF × LCM = Product of the two numbers. Therefore, LCM = 4320/12 = 360.

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Question (9)

QN0036480

Which prime factor must be present in the prime factorisation of a positive integer if its decimal representation ends in 0?

(A) 2 only
(B) 5 only
(C) Both 2 and 5
(D) 3 and 5

Correct Answer: C

Explanation:

A number ending in 0 is divisible by 10, and 10 = 2 × 5. Therefore, both 2 and 5 must occur as prime factors.

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Question (10)

QN0036481

The decimal expansion of a rational number depends, according to the chapter, on:

(A) only its numerator
(B) only its denominator's prime factorisation
(C) the difference between numerator and denominator
(D) the number of digits in the numerator

Correct Answer: B

Explanation:

The prime factorisation of the denominator reveals the nature of the decimal expansion of a rational number.

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Question (1)

Which statement correctly describes the Fundamental Theorem of Arithmetic?

Explanation:

Link:

Question (2)

Which of the following is the prime factorisation of 540?

Explanation:

Link:

Question (3)

If p is a prime number and p divides a², then which conclusion is necessarily true?

Explanation:

Link:

Question (4)

The prime factorisation of 840 is:

Explanation:

Link:

Question (5)

Which of the following is irrational?

Explanation:

Link:

Question (6)

Find the HCF of 84 and 126.

Explanation:

Link:

Question (7)

Find the LCM of 18 and 45.

Explanation:

Link:

Question (8)

If the HCF of two positive integers is 12 and their product is 4320, their LCM is:

Explanation:

Link:

Question (9)

Which prime factor must be present in the prime factorisation of a positive integer if its decimal representation ends in 0?

Explanation:

Link:

Question (10)

The decimal expansion of a rational number depends, according to the chapter, on:

Explanation:

Link:

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