MCQs on Real Numbers Class 10 Maths Chapter 1

Real Numbers Class 10 MCQs are available in this Maths article. It helps students practise important concepts from Chapter 1 of the CBSE Maths syllabus in an exam-oriented format. These multiple-choice questions with answers and detailed solutions cover topics such as Euclid’s Division Lemma, HCF and LCM, irrational numbers, decimal expansions, and the Fundamental Theorem of Arithmetic to strengthen conceptual understanding and improve problem-solving skills

MCQS on Chapter 1: Real Numbers for Class 10 With Answers

Question 1

Euclid's Division Lemma states that for two positive integers a and b, there exist unique integers q and r such that a = bq + r, where r must satisfy:

(A) 1 < r < b (B) 0 < r ≤ b

(C) 0 ≤ r < b  (D) 0 < r < b

Answer: C: 0 ≤ r < b

Explanation: The remainder r can be 0 but must always be less than b. Option A excludes r = 0. Option C (0 ≤ r < b) is correct.

Question 2

Using Euclid's Division Algorithm, the HCF of 231 and 396 is:

(A) 32 (B) 21

(C) 13 (D) 33 

Answer: D: 33 

Explanation: Using Euclid's Algorithm

396 = 231 × 1 + 165

231 = 165 × 1 + 66

165 = 66 × 2 + 33

66 = 33 × 2 + 0

Since remainder = 0, HCF = 33.

Question 3:

n² – 1 is divisible by 8, if n is:

(A) An integer (B) A natural number

(C) An odd integer (D) An even integer

Answer: C: An odd integer

Explanation: Let n = 2Q + 1 (odd). Then n² – 1 = (2Q+1)² – 1 = 4Q² + 4Q = 4Q(Q+1). Since one of Q or Q+1 is always even, the product Q(Q+1) is always even. So 4Q(Q+1) is always divisible by 8.

Question 4:

The least number that is divisible by all numbers from 1 to 5 is:

(A) 70 (B) 60

(C) 80 (D) 90

Answer: B: 60

Explanation We need LCM(1, 2, 3, 4, 5).

LCM = 2² × 3 × 5 = 4 × 3 × 5 = 60.

Question 5:

Which of the following is NOT irrational?

(A) 3 + √7 (B) 3 – √7

(C) (3 + √7)(3 – √7)  (D) 3√7

Answer: C: (3 + √7)(3 – √7) 

Explanation: (3 + √7)(3 – √7) = 3² – (√7)² = 9 – 7 = 2, which is rational. 

Question 6: 

The product of two irrational numbers is:

(A) Always irrational (B) Always rational

(C) May be rational or irrational (D) Always an integer

Answer: C: May be rational or irrational

Explanation: Counter-examples: √2 × √2 = 2 (rational), but √2 × √3 = √6 (irrational).

Question 7:

The decimal expansion of 22/7 is:

(A) Terminating  (B) Non-terminating and repeating 

(C) Non-terminating and non-repeating  (D) None of the above

Answer: B: Non-terminating and repeating 

Explanation: Its denominator is 7 and as a factor of 7, which is neither 2 nor 5. So it is non-terminating repeating: 22/7 = 3.142857142857…

Question 8:

If p/q is a rational number (q ≠ 0) and q = 2³ × 5², then p/q has a decimal expansion which terminates after:

(A) 2 places (B) 3 places 

(C) 5 places (D) 6 places

Answer: B: 3 places 

Explanation: q = 2³ × 5². To convert to /10ⁿ form, we multiply by 5/5 to make q = 2³ × 5³ = 10³. So the decimal terminates after 3 places (max(3, 2) = 3).

Question 9:

Three bells toll at intervals of 9, 12, and 15 minutes respectively. If they toll together at 10:00 AM, when will they next toll together?

(A) 10:45 AM (B) 11:00 AM

(C) 11:00 AM  (D) 11:30 AM

Answer: C: 11:00 AM 

Explanation: They toll together again after LCM(9,12,15) = 60 minutes. 10:00 AM + 60 min = 11:00 AM.

Question 10:

If HCF(a, b) = 12 and a × b = 1800, then LCM(a, b) is:

(A) 3600 (B) 900

(C) 150  (D) 90

Answer: C: 150 

Explanation: Using HCF × LCM = a × b:

12 × LCM = 1800

LCM = 1800 ÷ 12 = 150

Question 11: 

The largest number that divides 70 and 125, leaving remainders 5 and 8 respectively, is:

(A) 65 (B) 15

(C) 13 (D) 25

Answer: C: 13

Explanation: The required number divides (70 – 5) = 65 and (125 – 8) = 117 exactly. So the answer is HCF (65, 117) = 13

Question 12: 

If the HCF of 65 and 117 is expressible in the form 65m – 117, then the value of m is:

(A) 4 (B) 2 

(C) 1 (D) 3

Answer: B: 2

Explanation: HCF(65, 117) = 13. Now, 65m – 117 = 13 ⇒ 65m = 130 ⇒ m = 2.

Question 14:

The prime factorisation of 96 is:

(A) 2⁵ × 3 (B) 2⁶

(C) 2⁴ × 3 (D) 2⁴ × 3²

Answer: A: 2⁵ × 3

Explanation: 96 = 2 × 2 × 2 × 2 × 2 × 3 = 2⁵ × 3.

Question 15: 

5 – √3 is:

(A) A rational number (B) An irrational number 

(C) A whole number (D) A natural number

Answer: B: An irrational number 

Explanation: 5 is rational and √3 is irrational. Rational – Irrational = Irrational