Real Numbers — Class 10 MCQs with Answers
Class 10 CBSE Mathematics · Chapter 1
90 practice questions · 30 Easy · 30 Medium · 30 Hard · Updated
Practise the most important Class 10 CBSE Mathematics questions from Chapter 1, "Real Numbers". You get 9 timed quizzes made from 90 NCERT-based MCQs, with answers and explanations. The questions are split into 30 Easy, 30 Medium and 30 Hard. Warm up on the basics, then move on to the exam-level questions that set top scorers in CBSE Board exams and the JEE & NEET foundation years apart.
To score well in "Real Numbers", focus on fast problem-solving, formula recall and step-by-step working. Each MCQ here is timed and uses exam-style marking (+4 correct, −1 wrong, 0 skipped). This trains you to stay accurate under time pressure, as real papers need. Every question has a short explanation, so a wrong answer becomes a quick lesson. It is the fastest way to fix gaps before a test.
Use this chapter for focused revision. Start with the Easy set to check your basics on Real Numbers, then move to Medium and Hard to practise applying them. Your accuracy, streaks and XP save automatically. This chapter also adds to your overall Class 10 Mathematics mastery score. 14 sample questions are solved in full below, with the answer and a worked explanation. Sign in free to start practising.
Key concepts: Real Numbers (Class 10 Mathematics)
Every composite number breaks into primes in exactly one way, and that single fact settles HCF, LCM and the irrationality of numbers such as √2. This chapter also explains when a fraction's decimal expansion terminates and when it repeats forever.
- Fundamental Theorem of Arithmetic
- Every composite number can be written as a product of primes, and that factorisation is unique apart from the order of the factors.
- Prime factorisation
- Expressing a number as a product of primes, such as 140 = 2² × 5 × 7. It is the starting point for both HCF and LCM.
- HCF (Highest Common Factor)
- The largest number dividing two numbers exactly; found by taking the SMALLEST power of each common prime.
- LCM (Lowest Common Multiple)
- The smallest number both numbers divide into; found by taking the HIGHEST power of every prime appearing in either.
Real Numbers — important questions & MCQs with answers (Class 10 Mathematics)
14 solved questions from this chapter's difficulty levels, each with its answer and explanation. The other 76 are timed and scored when you sign in.
- Q1Easy
The HCF of 12 and 18 is:
A.6✓ CorrectB.3C.2D.12Answer: A. 6
Explanation: Prime factorise both: 12 = 2²×3 and 18 = 2×3². HCF uses the lowest power of each common prime, 2¹×3¹ = 6 — not the highest power, which would instead give the LCM (36).
- Q2Easy
Which of the following is irrational?
A.√4B.22/7C.0.333...D.√2✓ CorrectAnswer: D. √2
Explanation: √4 = 2, 22/7, and 0.333... = 1/3 are all ratios of integers, so they're rational. √2 cannot be written as p/q since 2 is not a perfect square, making it irrational.
- Q3Easy
Which fraction has a terminating decimal expansion?
A.1/6B.13/3125✓ CorrectC.3/7D.11/30Answer: B. 13/3125
Explanation: A p/q fraction (lowest terms) terminates only when q's prime factors are limited to 2 and/or 5. Since 3125 = 5⁵ contains no other prime, 13/3125 terminates; 1/6, 3/7 and 11/30 all carry a 3 or 7.
- Q4Easy
The LCM of 15, 25, and 75 is:
A.25B.75✓ CorrectC.150D.375Answer: B. 75
Explanation: Prime factorise: 15 = 3×5, 25 = 5², 75 = 3×5². LCM takes the highest power of each prime present, 3¹×5² = 75.
- Q5Easy
Which of the following is an irrational number?
A.22/7B.√16C.0.3̄D.√7✓ CorrectAnswer: D. √7
Explanation: √7 is irrational because 7 is not a perfect square, so its square root can't be written as p/q. √16 = 4, by contrast, is a whole number and clearly rational.
- Q6Easy
The decimal expansion of 17/8 is:
A.Non-terminating non-recurringB.Non-terminating recurringC.Terminating✓ CorrectD.Cannot be determinedAnswer: C. Terminating
Explanation: Denominator 8 = 2³ has only 2 as a prime factor, so the decimal terminates. Dividing out: 17/8 = 2.125.
- Q7Medium
If HCF(a, b) × LCM(a, b) = 1800 and a = 24, then b is:
A.75✓ CorrectB.60C.90D.120Answer: A. 75
Explanation: For two numbers, HCF(a,b) × LCM(a,b) = a × b. Here that product is already given as 1800, so 1800 = 24 × b, giving b = 75.
- Q8Medium
Which statement is TRUE?
A.Sum of two irrationals is always irrationalB.Product of a non-zero rational and an irrational is irrational✓ CorrectC.Quotient of two irrationals is always irrationalD.Square of an irrational is always irrationalAnswer: B. Product of a non-zero rational and an irrational is irrational
Explanation: A non-zero rational times an irrational is always irrational — provable directly. The other options fail on counterexamples: √2 + (−√2) = 0 and √8/√2 = 2 are both rational despite involving irrationals.
- Q9Medium
Which of these has a terminating decimal expansion?
A.7/30B.17/21C.64/455D.13/3125✓ CorrectAnswer: D. 13/3125
Explanation: A fraction in lowest terms terminates only if its denominator has no prime factors besides 2 and 5. 30, 21 and 455 all carry a 3, 7 or 13, but 3125 = 5⁵ is pure powers of 5, so 13/3125 terminates.
- Q10Medium
Three bells toll at intervals of 9, 12 and 15 minutes respectively. If they toll together at 7:00 AM, they will next toll together at:
A.7:45 AMB.8:00 AMC.10:00 AM✓ CorrectD.9:30 AMAnswer: C. 10:00 AM
Explanation: They next toll together after LCM(9, 12, 15) minutes. 9 = 3², 12 = 2²×3, 15 = 3×5, so LCM = 2²×3²×5 = 180 min = 3 h — added to 7:00 AM gives 10:00 AM.
- Q11Medium
Which of the following statements is always true?
A.Sum of two irrationals is irrationalB.Product of two irrationals is irrationalC.Product of a non-zero rational and an irrational is irrational✓ CorrectD.Difference of two irrationals is irrationalAnswer: C. Product of a non-zero rational and an irrational is irrational
Explanation: Non-zero rational × irrational is always irrational — provable directly. The others fail on counterexamples: √2 + (−√2) = 0, √2 × √2 = 2, and √2 − √2 = 0 are all rational despite involving irrationals.
- Q12Hard
The largest number which divides 70 and 125, leaving remainders 5 and 8 respectively, is:
A.65B.13✓ CorrectC.875D.1750Answer: B. 13
Explanation: Subtract the remainders first: the divisor must exactly divide 70 − 5 = 65 and 125 − 8 = 117. HCF(65, 117): 65 = 5×13, 117 = 9×13, so HCF = 13.
- Q13Hard
Which method is used to prove that √2 is irrational?
A.Mathematical inductionB.Direct constructionC.Contradiction (assume rational, derive contradiction)✓ CorrectD.Counter-exampleAnswer: C. Contradiction (assume rational, derive contradiction)
Explanation: The standard proof assumes √2 = p/q in lowest terms and derives that both p and q must be even, contradicting the 'lowest terms' assumption — this is proof by contradiction, not induction or a direct construction.
- Q14Hard
If 5/64 has a terminating decimal, the denominator after expansion will be of the form:
A.3ⁿ × 7ᵐB.2ⁿ × 5ᵐ✓ CorrectC.any primeD.only 10ⁿAnswer: B. 2ⁿ × 5ᵐ
Explanation: A terminating decimal's denominator (in lowest terms) is always of the form 2ⁿ×5ᵐ — no other prime is allowed. Here 64 = 2⁶ fits that form exactly, with m = 0.
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Start this chapter free →Real Numbers — FAQs
What are the key concepts in Class 10 Mathematics Real Numbers?+
Every composite number breaks into primes in exactly one way, and that single fact settles HCF, LCM and the irrationality of numbers such as √2. This chapter also explains when a fraction's decimal expansion terminates and when it repeats forever. Key ideas include Fundamental Theorem of Arithmetic, Prime factorisation, HCF (Highest Common Factor), LCM (Lowest Common Multiple).
What does Class 10 Mathematics Chapter 1 (Real Numbers) cover on XamBaaz?+
It has 90 NCERT-based MCQs on "Real Numbers": 30 Easy, 30 Medium and 30 Hard. Together they make 9 timed quizzes, and you never get the same set twice. Every question has an instant explanation. They help you prepare for CBSE Board exams and the JEE & NEET foundation years.
Are these "Real Numbers" questions free to practise?+
Yes. Sign in with Google to practise "Real Numbers" free. Full unlimited access is ₹999/year on a launch offer until 1 December 2026. No chapter is charged separately.
How should I revise "Real Numbers" for the exam?+
Start with the Easy quiz to check your basics, then try Medium and Hard to practise applying them. There are 9 timed quizzes on this chapter, so you can come back for a fresh set instead of one you have seen. Read each explanation, retry the questions you miss, and track your accuracy until it stays high.
Are these "Real Numbers" MCQs available with answers?+
Yes. 14 sample questions are shown here in full, each with the correct option and a step-by-step "Why" explanation. Sign in free with Google to start practising, with instant scoring.
Is there negative marking in the "Real Numbers" quizzes?+
Yes. The timed quizzes use exam-style marking: +4 for a right answer, −1 for a wrong one and 0 for a skip. MHT-CET and CBSE board papers have no negative marking. Our mocks for those are scored their way.
What are the important questions from Real Numbers (Class 10 Mathematics)?+
The questions that matter most test Fundamental Theorem of Arithmetic, Prime factorisation, HCF (Highest Common Factor), LCM (Lowest Common Multiple). This page shows 14 solved important MCQs with answers and explanations; all 90 questions on the chapter are available as timed quizzes once you sign in.
Is there an online quiz for Real Numbers?+
Yes — Class 10 Mathematics Real Numbers has timed online quizzes at Easy, Medium and Hard levels, with instant scoring and a worked explanation on every question. The first quiz on the chapter is free.