CBSE Important Questions for Class 10 Maths (2025-26) Chapter 1: Real Numbers with Answers, Download PDF

CBSE Class 10 Mathematics begins with Chapter 1: Real Numbers, one of the most important and scoring topics in the syllabus. This chapter lays the foundation for advanced mathematical concepts such as arithmetic progressions, polynomials, and number theory. It mainly covers topics like the Euclid’s Division Lemma, Fundamental Theorem of Arithmetic, HCF and LCM, and Irrational Numbers. Real Numbers is a completely scoreable chapter that carries six marks if correctly solved. Students can download the pdf for free to keep and practice in future. 

For students preparing for the CBSE Class 10 Board Exam 2026, mastering Real Numbers is crucial as it carries a good weightage and often includes both short and long-answer questions. Below are some important questions with detailed answers to help you strengthen your concepts and prepare effectively.

CBSE Class 10 Maths Exam 2026: Key Highlights

Important Questions and Answers for CBSE Class 10 Maths Chapter 1: Real Numbers

Section A: One-mark Questions

1. Which of the following is an irrational number?
   a) 0.75 b) 7/25 c) √5 d) 4.2

2. The prime factorization of 84 is
   a) 2² × 3 × 7  b) 2 × 3² × 7  c) 2² × 7²  d) 2³ × 3 × 7

3. If HCF of two numbers is 12 and their product is 864, then their LCM is
   a) 36  b) 72  c) 144  d) 576

4. Which of these pairs are co-prime?
   a) (8, 12)  b) (9, 28)  c) (14, 21)  d) (6, 9)

5. What is the HCF of 91 and 287?

6. Write the representation of 60 as a product of prime factors.

7. State Euclid’s Division Lemma (in one line).

8. What is the nth term of Euclid’s lemma expression? Or: In Euclid’s lemma, what is r in “a = bq + r”?

9. Is 0 an irrational number or rational number?

10. What is the HCF of 0 and any nonzero integer a?

Section B: Two-mark Questions

1. Use Euclid’s algorithm (division method) to find HCF of 1078 and 462.

2. Show that √7 is irrational.

3. Find LCM and HCF of 240 and 360 using prime factorization.

4. Express 2310 as a product of its prime factors.

5. If two numbers are 45 and 60, find their HCF and LCM.

6. If p is prime, show √p is irrational.

7. Using Euclid’s Division Lemma, find HCF of 1234 and 56.

8. Prove that there are infinitely many primes.

9. Show that between any two rational numbers there exists an irrational number (or briefly argue irrational numbers are “dense”).

10. Using prime factorization, find LCM of 84, 90, and 126.

Section C: Three-mark Questions

1. Prove that √2 + √3 is irrational (or show that the sum of two irrational square roots in some cases is irrational).

2. Use Euclid’s Division Lemma to derive that HCF (a, b) = HCF(b, r) in the division step.

3. Find HCF and LCM of 450, 600, 675.

4. If the HCF of two numbers is 14 and their LCM is 1680, and one number is 84, find the other.

5. Prove that √3 + √5 is irrational.

6. If 2^m = 3^n for some positive integers m, n, show that this leads to a contradiction (i.e., no nonzero integer solution).

7. Let a and b be two positive integers such that a divides b. Prove that HCF(a, b) = a and LCM(a, b) = b.

8. If p, q, r are three primes (not necessarily distinct), prove that every integer N > 1 can be written uniquely (except for ordering) as a product of primes.

9. Prove that √(p/q) is irrational if p and q are positive integers and p/q is not a perfect square.

10. Show that among any n + 1 positive integers, there exist two whose difference is divisible by n. (This is a bit more general, but ties into the divisibility idea.)

Section D: Case Study Based Questions (4 Marks)

Case Study 1
A teacher gives three students numbers 462, 792, and 1378 and asks them to find their HCF and LCM, and to check whether their square roots are rational or irrational.

• (a) Find HCF and LCM of 462, 792, and 1378.

• (b) Decide which of √462, √792, √1378 are irrational.

• (c) Express each of the three numbers in prime factorization form.

• (d) Are any two of the numbers co-prime? Justify.

Case Study 2
A factory has three machines producing well counts of items: 168, 210, and 280 pieces per hour. The manager wants to find the largest batch size that divides all production counts evenly and also find how many such batches per hour.

• (a) What is the largest batch size that divides 168, 210, 280 evenly (i.e. HCF)?

• (b) How many batches per hour for each machine?

• (c) Also find the least number of hours after which all three machines produce a common multiple of batch size (i.e. LCM and its interpretation).

Case Study 3
A competition has prizes for prime numbered participants. After the event, the organizer realized that some participants have repeated prime factors in their registration codes. He lists codes: 462, 770, 1001, 1210

• (a) Factorize each code into primes.

• (b) Among those codes, find any two which are co-prime.

• (c) What is the HCF and LCM among all four codes?

• (d) Are the square roots of any of those codes rational or irrational?

Case Study 4
In a number theory research group, the numbers 123, 246, 369 are being considered. The researcher wants to explore their common divisors, possible irrational square roots, and prime factorizations.

• (a) Factorize 123, 246, 369.

• (b) Find HCF and LCM.

• (c) Which of √123, √246, √369 are irrational?

• (d) Are any pairs co-prime?

Case Study 5
A software algorithm is given three integer inputs: 225, 360, 450. The algorithm must compute greatest common divisor, least common multiple, factorization, and test for square roots being rational or irrational.

• (a) Prime factorize 225, 360, 450.

• (b) Compute HCF and LCM.

• (c) Are √225, √360, √450 rational or irrational?

• (d) Are any of the numbers co-prime pairs?

Answers

Section A: One Marks

To Check all the detailed answers of the important question of CBSE Class 10 Maths Chapter 1: Real Numbers, Dowload PDF from the link provided below:

Why Are These Questions Important?

• Euclid’s Lemma / Division Algorithm and HCF/LCM problems appear very frequently in CBSE exams.

• Proofs of irrationality (√2, √p, sums of square roots) are standard.

• Prime factorization is a recurring subtopic.

• Case-study / application-based questions are becoming popular in newer assessments to test understanding in context.

• Mixed MCQs in one-mark sections help test quick recall and concept clarity.

The CBSE Class 10 Chapter 1: Real Numbers is essential for building strong mathematical reasoning skills. Questions from this chapter test students’ conceptual understanding and problem-solving accuracy. Regular practice of important questions like these ensures full marks in short and long answer sections. Students are advised to revise the Euclid’s Lemma, prime factorization method, and properties of irrational numbers thoroughly before the board exam. Keep solving sample papers and NCERT exemplar questions for better confidence and accuracy.

Also Check:

CBSE Class 10 Maths Syllabus 2025-26

CBSE Class 10 Maths 5 Month Study Plan for Board Exam 2026

CBSE Class 10 Maths Half Yearly Sample Paper 2025 with Solution (Standard)