Chapter 1: Real Numbers

Class 10 Mathematics · MCQs with answers

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What is the prime factorisation of 140?
1. (A) 2 × 5 × 14
2. (B) 2 × 7 × 10
3. (C) 2² × 35
4. (D) 2² × 5 × 7
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Correct answer: (D) 2² × 5 × 7

Explanation: 140 = 2 × 70 = 2 × 2 × 35 = 2² × 5 × 7. In a prime factorisation every factor must be prime; 14, 10 and 35 are not prime.

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What is the HCF of 96 and 404?
1. (A) 2
2. (B) 4
3. (C) 8
4. (D) 12
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Correct answer: (B) 4

Explanation: 96 = 2⁵ × 3 and 404 = 2² × 101. The common factor is 2², so HCF = 4.

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What is the LCM of 12, 15 and 21?
1. (A) 3780
2. (B) 1260
3. (C) 420
4. (D) 210
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Correct answer: (C) 420

Explanation: 12 = 2² × 3, 15 = 3 × 5, 21 = 3 × 7. Taking the highest power of each prime, LCM = 2² × 3 × 5 × 7 = 420.

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The HCF of two numbers is 9 and their product is 1620. What is their LCM?
1. (A) 90
2. (B) 162
3. (C) 180
4. (D) 360
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Correct answer: (C) 180

Explanation: HCF × LCM = product of the two numbers. So LCM = 1620 ÷ 9 = 180.

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What does the Fundamental Theorem of Arithmetic state?
1. (A) Every number is even
2. (B) Every composite number can be written as a product of primes in exactly one way (apart from order)
3. (C) The sum of two primes is always prime
4. (D) Every prime number is odd
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Correct answer: (B) Every composite number can be written as a product of primes in exactly one way (apart from order)

Explanation: By the Fundamental Theorem, every composite number has a unique prime factorisation; only the order of the factors can change.

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Which of these is an irrational number?
1. (A) √2
2. (B) √4
3. (C) 0.25
4. (D) 3/7
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Correct answer: (A) √2

Explanation: √2 cannot be written as p/q (q ≠ 0), so it is irrational. √4 = 2, 0.25 = 1/4 and 3/7 are rational.

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What kind of number is 3 + 2√5?
1. (A) Rational
2. (B) An integer
3. (C) Irrational
4. (D) A natural number
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Correct answer: (C) Irrational

Explanation: If 3 + 2√5 were rational, then √5 = (number − 3)/2 would also be rational, which is false. So 3 + 2√5 is irrational.

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The product of a non-zero rational number and an irrational number is always:
1. (A) Rational
2. (B) Irrational
3. (C) Zero
4. (D) An integer
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Correct answer: (B) Irrational

Explanation: For example, 2 × √3 = 2√3 is irrational. If the product were rational, the irrational number = product ÷ rational number would be rational too, which is impossible.

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For which natural number n does 4ⁿ end with the digit 0?
1. (A) n = 5
2. (B) n = 10
3. (C) n = 20
4. (D) For no value of n
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Correct answer: (D) For no value of n

Explanation: To end in 0, a number's prime factorisation must contain both 2 and 5. 4ⁿ = 2²ⁿ never contains 5, so 4ⁿ never ends in 0.

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What is the HCF of two co-prime numbers?
1. (A) 0
2. (B) Their product
3. (C) 1
4. (D) The smaller number
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Correct answer: (C) 1

Explanation: Co-prime numbers have no common factor other than 1, so their HCF is 1 and their LCM equals their product.

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What is the HCF of 26 and 91?
1. (A) 91
2. (B) 26
3. (C) 13
4. (D) 7
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Correct answer: (C) 13

Explanation: 26 = 2 × 13 and 91 = 7 × 13. The common factor is 13, so HCF = 13. (LCM = 2 × 7 × 13 = 182.)

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What is the smallest number exactly divisible by both 306 and 657?
1. (A) 9
2. (B) 2178
3. (C) 22338
4. (D) 201042
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Correct answer: (C) 22338

Explanation: This is the LCM. 306 = 2 × 3² × 17 and 657 = 3² × 73, so LCM = 2 × 3² × 17 × 73 = 22338.

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What is the power of 2 in the prime factorisation of 144?
1. (A) 6
2. (B) 4
3. (C) 3
4. (D) 2
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Correct answer: (B) 4

Explanation: 144 = 16 × 9 = 2⁴ × 3². So the power of 2 is 4.

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Which of these is a rational number?
1. (A) √9
2. (B) √2
3. (C) π
4. (D) √3 + 1
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Correct answer: (A) √9

Explanation: √9 = 3 = 3/1, which is of the form p/q, so it is rational. √2, π and √3 + 1 are irrational.

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The rule "HCF × LCM = product of the numbers" always holds for:
1. (A) Any three positive integers
2. (B) Prime numbers only
3. (C) Any two positive integers
4. (D) Never
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Correct answer: (C) Any two positive integers

Explanation: The rule always holds for two numbers. It need not hold for three: for 2, 3 and 4, HCF = 1 and LCM = 12, but the product is 24.

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What is the HCF of 8, 9 and 25?
1. (A) 1
2. (B) 2
3. (C) 5
4. (D) 1800
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Correct answer: (A) 1

Explanation: 8 = 2³, 9 = 3² and 25 = 5². They share no prime factor, so HCF = 1. (Their LCM is 1800.)

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Sonia takes 18 minutes and Ravi takes 12 minutes to go once round a field. They start at the same point and time and go in the same direction. After how many minutes will they meet again at the starting point?
1. (A) 216
2. (B) 36
3. (C) 30
4. (D) 6
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Correct answer: (B) 36

Explanation: They meet at a time that is a multiple of both 18 and 12, which is the LCM. 18 = 2 × 3² and 12 = 2² × 3, so LCM = 2² × 3² = 36 minutes.

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What is the largest number that divides 70 and 125, leaving remainders 5 and 8 respectively?
1. (A) 5
2. (B) 8
3. (C) 13
4. (D) 65
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Correct answer: (C) 13

Explanation: The number must divide 70 − 5 = 65 and 125 − 8 = 117 exactly. 65 = 5 × 13 and 117 = 3² × 13, so HCF = 13.

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What is the prime factorisation of 3825?
1. (A) 3² × 5² × 17
2. (B) 3 × 5² × 51
3. (C) 3² × 5 × 85
4. (D) 3³ × 5² × 7
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Correct answer: (A) 3² × 5² × 17

Explanation: 3825 = 25 × 153 = 5² × 9 × 17 = 3² × 5² × 17. Among the other choices, 51 and 85 are not prime, and 3³ × 5² × 7 = 4725.

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What kind of number is 7 × 11 × 13 + 13?
1. (A) Prime
2. (B) Neither prime nor composite
3. (C) Irrational
4. (D) Composite
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Correct answer: (D) Composite

Explanation: 7 × 11 × 13 + 13 = 13 × (77 + 1) = 13 × 78. It has factors other than 1 and itself, so it is composite.

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By Euclid's division lemma, for positive integers a and b, a = bq + r. What condition does r satisfy?
1. (A) 0 < r ≤ b
2. (B) 0 ≤ r < b
3. (C) r > b
4. (D) r = b
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Correct answer: (B) 0 ≤ r < b

Explanation: The remainder r is always at least 0 and smaller than the divisor b, that is, 0 ≤ r < b.

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Using Euclid's division algorithm, what is the HCF of 135 and 225?
1. (A) 15
2. (B) 45
3. (C) 75
4. (D) 135
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Correct answer: (B) 45

Explanation: 225 = 135 × 1 + 90; 135 = 90 × 1 + 45; 90 = 45 × 2 + 0. When the remainder becomes 0, the divisor, 45, is the HCF.

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Without long division, which of these rational numbers has a terminating decimal expansion?
1. (A) 17/6
2. (B) 64/455
3. (C) 29/343
4. (D) 13/3125
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Correct answer: (D) 13/3125

Explanation: In lowest terms, the expansion terminates if the denominator's prime factors are only 2s and/or 5s. 3125 = 5⁵, while 6, 455 and 343 have factors such as 3, 7 or 13.

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After how many decimal places does the decimal expansion of 17/8 terminate?
1. (A) 1
2. (B) 2
3. (C) 3
4. (D) 4
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Correct answer: (C) 3

Explanation: 17/8 = 17 × 125 / (8 × 125) = 2125/1000 = 2.125. The denominator is 2³, so the expansion ends after 3 decimal places.

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Assertion (A): 6ⁿ cannot end with the digit 0 for any natural number n. Reason (R): A number ends in 0 only if its prime factorisation contains both 2 and 5.
1. (A) Both A and R are true, and R is the correct explanation of A.
2. (B) Both A and R are true, but R is not the correct explanation of A.
3. (C) A is true, but R is false.
4. (D) A is false, but R is true.
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Correct answer: (A) Both A and R are true, and R is the correct explanation of A.

Explanation: 6ⁿ = 2ⁿ × 3ⁿ never contains 5, so it cannot end in 0. R correctly explains A.

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