Real Numbers Questions Class 10 2026-27

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Real Numbers is one of the most important chapters in Class 10 Maths. It builds the foundation for concepts like HCF, LCM, Euclid’s Division Lemma, and irrational numbers. In this post, you will find important MCQs, short answer questions, and case study-based questions for better exam preparation.

Questions

Section A: MCQs

  1. The product of a non-zero rational number and an irrational number is
    a) Always irrational
    b) always rational
    c) rational or irrational
    d) none
  2. The ratio of HCF to LCM of the least composite number and the least prime number is:
    a) 1:2
    b) 2:1
    c) 1:1
    d) 1:3
  3. If ‘p’ and ‘q’ are natural numbers and ‘p’ is a multiple of ‘q’, then what is the HCF of ‘p’ and ‘q’?
    a) pq
    b) p
    c) q
    d) p + q
  4. If a and b are two coprime numbers, then a³ and b³ are
    a) Coprime
    b) not coprime
    c) even
    d) odd
  5. The exponent of 5 in the prime factorisation of 3750 is
    a) 3
    b) 4
    c) 5
    d) 6
  6. What is the greatest possible speed at which a girl can walk 95 m and 171 m in an exact number of minutes?
    a) 17 m/min
    b) 19 m/min
    c) 23 m/min
    d) 13 m/min
  7. If a = x³y² and b = xy³, where x, y are prime numbers, then the result obtained by dividing the product by LCM (a, b) is
    a) xy
    b) xy²
    c) x³y³
    d) x²y²
  8. If n is any natural number, then 6ⁿ − 5ⁿ always ends with which digit?
    a) 0
    b) 1
    c) 2
    d) 3

Section B: Short Answer

  1. Find the sum of exponents of prime factors in the prime factorization of 196.
  2. If HCF of 144 and 180 is expressed in the form 13m – 16, find m.
  3. What is the remainder when the square of any prime number greater than 3 is divided by 6?
  4. Find the least number divisible by all numbers from 1 to 10.
  5. If a and b are prime numbers, what is their LCM?
  6. If n = 2³ × 3⁴ × 5⁴ × 7, find number of trailing zeros in n.
  7. If LCM of a and 18 is 36 and HCF is 2, find a.
  8. If 3 is least prime factor of a and 7 of b, find least prime factor of a + b.

🔹 Section C: Long Answer

  1. Show that (12)ⁿ cannot end with digit 0 or 5.
  2. Can two numbers have HCF = 16 and LCM = 380? Give reason.
  3. Explain whether 3 × 12 × 101 + 4 is prime or composite.
  4. Three planks of lengths 63 m, 42 m, and 49 m are cut into equal pieces. Find least number of planks.
  5. Three bells ring at intervals of 9, 12, 15 minutes. When will they ring together again?
  6. Check whether 4ⁿ can end with digit 0.
  7. LCM is 14 times HCF and sum is 600. One number is 280. Find the other.
  8. Find HCF and LCM of 96 and 404.
  9. Find greatest number dividing 398, 436, 542 leaving remainders 7, 11, 15.
  10. Find maximum capacity of container measuring 850 L and 680 L exactly.

Section D: Advanced / Case Study

  1. Traffic lights change after 48s, 72s, 108s. When will they coincide again?
  2. Prove √8 is irrational.
  3. Prove √2 + √5 is irrational.
  4. Prove (3 + 2√5)² is irrational.
  5. 2002 cartons of Lassi and 2618 cartons of Frooti are stacked. Find maximum cartons per stack.
  6. Find largest 4-digit number divisible by 12, 15, and 24.
  7. Numbers are in ratio 5:3 and LCM = 90. Find HCF.
  8. If HCF = 120 and LCM = 252k, find k.
  9. Find smallest number which when increased by 17 is divisible by 520 and 468.
  10. Find smallest number leaving remainder 7 when divided by 35, 56, 91.
  11. Find minimum number of square tiles for courtyard 18m 72cm × 13m 20cm.

🔹 Assertion Reason

  1. A: HCF (336, 54) = 6, LCM = 3000
    R: Sum of exponents in 196 is 4
  2. A: HCF = product of smallest powers of common primes
    R: HCF (12, 21, 15) = 3
  3. A: 6ⁿ ends with 0
    R: Numbers ending with 0 have factors 2 and 5

🔹 Case Study

  1. Section A has 32 students and B has 36 students
  2. Minimum books needed = ?
  3. Express 36 as product of primes
  4. 7×11×13×15 + 15 is ______ number
  5. If p = ab² and q = a²b, find LCM

Boost your practice

👉 Check Class 10 Polynomials MCQs for algebra practice
👉 Practice more with Pair of Linear Equations Questions
👉 Improve concepts with Quadratic Equations Important Questions
👉 Revise basics using Arithmetic Progressions Questions
👉 Strengthen geometry via Triangles MCQs
👉 Explore coordinate-based problems in Coordinate Geometry Questions

🎯 Solutions

MCQs Answers

1-a, 2-d, 3-c, 4-a, 5-b, 6-b, 7-d, 8-b

Short Answers

  1. 196 = 2² × 7² → sum = 4
  2. HCF = 36 → 13m –16 = 36 → m = 4
  3. Remainder = 1
  4. LCM (1–10) = 2520
  5. LCM = ab
  6. Zeros = min(3,4) = 3
  7. a = 4
  8. Least prime factor = 2

Long Answers

  1. Ends with 2,4,6,8 → never 0 or 5
  2. Not possible (HCF×LCM ≠ product)
  3. Composite
  4. HCF = 7 → 9 planks
  5. LCM = 180 min
  6. No
  7. Other number = 12
  8. HCF = 4, LCM = 9696
  9. Answer = 17
  10. Answer = 170

Advanced

  1. LCM = 432 sec → 7:07:12 AM
    28–30. Irrational (proof by contradiction)
  2. HCF = 154
  3. 9960
  4. HCF = 6
  5. k = 50
  6. Answer = 2583
  7. Answer = 287
  8. Tiles = 90

Case Study

  1. LCM (32,36) = 288
  2. 2² × 3²
  3. Composite
  4. LCM = a²b²

❓ FAQs

Q1. What are real numbers?

All rational and irrational numbers together form real numbers.

Q2. What is HCF?

HCF is the highest common factor of two numbers.

Q3. What is LCM?

LCM is the smallest multiple common to numbers.

Q4. Is √2 rational?

No, √2 is irrational.