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Class 10 Real Numbers: 8 practice questions with answers, step by step
These are 8 practice questions on NCERT Class 10 Maths Chapter 1, Real Numbers, each with a worked answer. They cover what the chapter asks: HCF and LCM by prime factorisation, the product rule for two numbers, a word problem, why that rule fails for three numbers, and the first step of an irrationality proof. Try each one on paper before you open the answer. The explanations behind the method are in our guide to NCERT Class 10 Chapter 1, Real Numbers. Testing a square by prime factorisation is the topic of Class 8 A Square and A Cube, the same idea two classes earlier.
These questions are our own, drafted with AI help. Every answer was re-computed by a test script, but no practising teacher has reviewed them yet. If you find a mistake, please tell us.
What the chapter says you need
Facts from the NCERT textbook (Mathematics, Textbook for Class X, Reprint 2026-27, Chapter 1, printed pages 1 to 9):
- Fundamental Theorem of Arithmetic (Theorem 1.1, page 3): every composite number can be expressed as a product of primes, and this factorisation is unique, apart from the order of the prime factors.
- HCF and LCM by prime factorisation (page 4): the HCF is the product of the smallest power of each common prime factor; the LCM is the product of the greatest power of each prime factor involved in the numbers.
- For any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b (page 4). For three numbers the product of the numbers is not equal to the product of their HCF and LCM (remark, page 5).
- Theorem 1.2 (page 6): if a prime p divides a², then p divides a, where a is a positive integer. It is the first step in proving that √2 and √3 are irrational (summary, page 9).
HCF and LCM
Question 1. Find the LCM of 12 and 18.
Answer: 36. 12 = 2² × 3 and 18 = 2 × 3². Take the greatest power of each prime: LCM = 2² × 3² = 36.
Question 2. Find the HCF and the LCM of 24 and 36 by prime factorisation, and check that HCF × LCM = 24 × 36.
Answer: HCF 12, LCM 72. 24 = 2³ × 3 and 36 = 2² × 3². HCF = smallest power of each common prime = 2² × 3 = 12. LCM = greatest power of each prime = 2³ × 3² = 72. Check: 12 × 72 = 864 and 24 × 36 = 864.
Question 3. The product of the HCF and the LCM of two numbers is 1,080. One of the numbers is 24. Find the other.
Answer: 45. HCF × LCM = the product of the two numbers (page 4), so the other number = 1,080 ÷ 24 = 45.
Question 4. Two school bells ring every 12 minutes and every 20 minutes. They ring together at 8:00. After how many minutes will they next ring together?
Answer: 60 minutes. They ring together again after the LCM of 12 and 20. 12 = 2² × 3 and 20 = 2² × 5, so LCM = 2² × 3 × 5 = 60 minutes (at 9:00).
Prime factorisation and three numbers
Question 5. Write 360 as a product of primes. How many different primes does it have, and which is the largest?
Answer: 360 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5. It has 3 different primes (2, 3 and 5), and the largest is 5. By Theorem 1.1 this is the only way to write it, apart from the order.
Question 6. Find the HCF and the LCM of 12, 18 and 24. Is HCF × LCM equal to 12 × 18 × 24?
Answer: HCF 6, LCM 72; no. 12 = 2² × 3, 18 = 2 × 3² and 24 = 2³ × 3. HCF = 2 × 3 = 6; LCM = 2³ × 3² = 72. So HCF × LCM = 6 × 72 = 432, but 12 × 18 × 24 = 5,184. They are not equal: the product rule holds for two numbers only. (Remark, page 5.)
Question 7. Can 15ⁿ end with the digit 0 for any natural number n?
Answer: no. A number ends in 0 only if it is divisible by 10 = 2 × 5, so both 2 and 5 would have to be prime factors. 15ⁿ = (3 × 5)ⁿ has only the primes 3 and 5. By the uniqueness in the Fundamental Theorem of Arithmetic (page 3), no other prime can appear, so 15ⁿ never ends with 0.
Irrational numbers
Question 8. A student begins to show that √3 is irrational by writing √3 = a/b with a and b coprime. Squaring gives 3b² = a². What does this tell you about a, and which result do you use?
Answer: 3 divides a. From 3b² = a², the prime 3 divides a². By Theorem 1.2 (page 6), if a prime divides a², it divides a, so 3 divides a. The proof then continues by writing a = 3c and showing that 3 also divides b, which contradicts a and b being coprime. (The same argument for √2 is Theorem 1.3, pages 6 and 7.)
Practise more, and check your marks
- Free practice, marked instantly in your browser: Class 10 Real Numbers questions. Questions 1, 3 and 4 above are in this set.
- Printable worksheets with answer keys, in English, Hindi and Marathi: Class 10 worksheets.
- See which unit this chapter belongs to and how CBSE splits the 80 theory marks in our Class 10 Maths syllabus 2026-27 guide.
- The same step-by-step style for the next chapter: Class 10 Polynomials.
Where CMDLearn fits
CMDLearn is building an AI tutor for Classes 1 to 10 that explains a step and checks the child's working, but leaves the writing to the child. It is still in development, and no child uses it yet. If you would like to hear when there is something to try, you can join the waitlist.
Sources
All opened and read on 21 October 2026. Page numbers are the printed page numbers of the textbook.
- NCERT, Mathematics, Textbook for Class X, Chapter 1 "Real Numbers" (Reprint 2026-27), printed pages 1-9 (Fundamental Theorem of Arithmetic, Theorem 1.1, page 3; HCF and LCM by prime factorisation and the product rule for two numbers, page 4; the remark on three numbers, page 5; Theorem 1.2 and the proof that √2 is irrational, pages 6-7; summary, page 9): ncert.nic.in/textbook/pdf/jemh101.pdf
The theorems and rules are from the textbook pages above. The questions, worked answers and checks are our own, and every number was re-computed by script. This guide is general information; follow your teacher and the official CBSE notices.