Mathematics becomes much easier when the basics are clear, and that is exactly why Real Numbers Class 10 MCQs are important for students preparing for CBSE board exams, school tests, and competency-based assessments. Chapter 1 builds the foundation for many upcoming maths concepts, so understanding topics like Euclid’s Division Lemma, irrational numbers, HCF, LCM, and decimal expansion is very important. Many students directly jump into solving questions without understanding the logic behind the chapter, and that’s where mistakes usually happen. Practicing Class 10 Maths Chapter 1 MCQs helps improve speed, accuracy, and conceptual clarity at the same time. This page is specially designed for students who want quick revision, exam-focused preparation, and better understanding of important concepts from the chapter. Along with objective questions, students should also focus on logic-based and competency-oriented patterns because CBSE is now focusing more on concept application instead of direct memorization.
Important Topics Covered in Real Numbers MCQs
Before solving the questions, students should revise these important concepts carefully because most Real Numbers objective questions are created from these areas.
- Euclid’s Division Lemma
- Euclid’s Division Algorithm
- Finding HCF using division method
- Fundamental Theorem of Arithmetic
- Prime factorization method
- Relationship between HCF and LCM
- Rational numbers
- Irrational numbers
- Decimal expansion of rational numbers
- Terminating decimals
- Non-terminating recurring decimals
- Coprime numbers
These topics form the complete base of CBSE Class 10 Real Numbers MCQs and are repeatedly asked in school exams and board-level practice papers.
Why Real Numbers Chapter is Important for Class 10 Boards
The chapter may look small at first, but its weightage and conceptual importance are quite high. Questions from this chapter are frequently asked in:
- Objective examinations
- Competency-based assessments
- Case study questions
- Assertion and reason format
- Internal school exams
- CBSE sample papers
Students who understand this chapter properly usually perform better in Algebra and Number System concepts later. Many board exam questions are designed in a tricky way where concepts of prime factorization and decimal expansion are mixed together. So regular practice of Real Numbers MCQ with answers becomes very important for scoring well.
Real Numbers Class 10 MCQs with Answers
Practice the most important and exam-focused Real Numbers Class 10 MCQs designed according to the latest CBSE pattern. These multiple choice questions will help students improve conceptual understanding, accuracy, and problem-solving speed for board exams, school tests, and competency-based assessments.
Q. Given two positive integers ‘a’ and ‘b’, which of the following statements correctly represents Euclid’s Division Lemma?
A) a = bq + r, where r > b
B) a = bq + r, where zero less than or equal to r less than b
C) b = aq + r, where zero less than r less than a
D) a = bq - r, where zero less than or equal to r less than b
Answer: B
Explanation: Euclid’s Division Lemma states that for positive integers a and b, a = bq + r where zero less than or equal to r less than b.
Q. Using Euclid’s Division Algorithm, find the HCF of one hundred ninety-six and thirty-eight thousand two hundred twenty.
A) 196
B) 38220
C) 98
D) 1
Answer: A
Explanation: Thirty-eight thousand two hundred twenty is divisible by one hundred ninety-six, so the HCF is 196.
Q. Which of the following statements best describes the Fundamental Theorem of Arithmetic?
A) Every even number can be expressed as a product of prime numbers
B) Every composite number can be expressed as a product of prime numbers in a unique way, except for the order of the prime factors
C) Every prime number can be expressed as a product of composite numbers
D) Every integer can be expressed as a sum of prime numbers
Answer: B
Explanation: The theorem states that every composite number has a unique prime factorization apart from the order of factors.
Q. If the HCF of two numbers is eighteen and their product is twelve thousand nine hundred sixty, what is their LCM?
A) 720
B) 233280
C) 180
D) 360
Answer: A
Explanation: Product of two numbers = HCF × LCM. Therefore, LCM = 12960 ÷ 18 = 720.
Q. Which of the following numbers has a non-terminating recurring decimal expansion?
A) 17/8
B) 64/455
C) 15/1600
D) 23/200
Answer: B
Explanation: Denominator 455 contains prime factor seven apart from two and five, so its decimal expansion is non-terminating recurring.
Q. Consider the number seven × eleven × thirteen + thirteen. Is it a prime number or a composite number?
A) Prime, because it is an odd number
B) Composite, because it can be expressed as a product of two numbers other than one and itself
C) Prime, because it is not divisible by two
D) Neither prime nor composite
Answer: B
Explanation: Taking thirteen common gives 13(7 × 11 + 1), so the number has factors other than one and itself.
Q. If p and q are two positive integers such that p = a²b³ and q = a³b, where a and b are prime numbers, then find the HCF(p, q).
A) a³b³
B) a²b
C) ab
D) a⁵b⁴
Answer: B
Explanation: HCF takes the smaller powers of common prime factors. Smaller powers are a² and b¹.
Q. The decimal expansion of the rational number fourteen thousand five hundred eighty-seven divided by one thousand two hundred fifty will terminate after how many decimal places?
A) One
B) Two
C) Three
D) Four
Answer: D
Explanation: 1250 = 2 × 5⁴. Highest power of five is four, so decimal terminates after four places.
Q. If LCM(96, 404) = 9696, then HCF(96, 404) is:
A) 4
B) 2
C) 8
D) 1
Answer: A
Explanation: HCF × LCM = Product of numbers. HCF = (96 × 404) ÷ 9696 = 4.
Q. Which of the following is an irrational number?
A) √4
B) 0.333...
C) √7
D) 22/7
Answer: C
Explanation: √7 cannot be expressed in the form p/q, so it is irrational.
Q. What is the HCF of the smallest prime number and the smallest composite number?
A) 1
B) 2
C) 3
D) 4
Answer: B
Explanation: Smallest prime number is two and smallest composite number is four. Their HCF is two.
Q. The product of a non-zero rational number and an irrational number is always:
A) Rational
B) Irrational
C) Integer
D) Whole number
Answer: B
Explanation: Multiplying a non-zero rational number with an irrational number always gives an irrational number.
Q. What is the maximum number of decimal places after which the decimal expansion of six divided by one thousand two hundred fifty will terminate?
A) 1
B) 2
C) 3
D) 4
Answer: D
Explanation: 1250 = 2 × 5⁴, so the decimal terminates after four decimal places.
Q. If ‘n’ is any natural number, then 6ⁿ can end with the digit:
A) 0
B) 5
C) 6
D) Any even digit
Answer: C
Explanation: Any power of six always ends with the digit six.
Q. According to Euclid’s Division Lemma, for any positive integer ‘a’ and ‘b = 3’, the possible values of remainder ‘r’ are:
A) 0, 1, 2, 3
B) 0, 1, 2
C) 1, 2, 3
D) Only 0
Answer: B
Explanation: Remainder must satisfy zero less than or equal to r less than 3.
Q. The sum of a rational and an irrational number is always:
A) Rational
B) Irrational
C) Integer
D) Could be rational or irrational
Answer: B
Explanation: Adding a rational number to an irrational number always results in an irrational number.
Q. The decimal expansion of thirteen divided by three thousand one hundred twenty-five is:
A) Terminating
B) Non-terminating recurring
C) Non-terminating non-recurring
D) Cannot be determined
Answer: A
Explanation: 3125 = 5⁵, so the decimal expansion terminates.
Q. If ‘a’ and ‘b’ are two co-prime numbers, then HCF(a, b) is:
A) a
B) b
C) ab
D) 1
Answer: D
Explanation: Co-prime numbers always have HCF equal to one.
Q. The least number that is divisible by all numbers from one to ten both inclusive is:
A) 100
B) 2520
C) 5040
D) 720
Answer: B
Explanation: The LCM of numbers from one to ten is 2520.
Q. Which of the following numbers is an irrational number?
A) 0.1010010001...
B) √9
C) 0.375
D) 5/2
Answer: A
Explanation: The decimal expansion is non-terminating and non-repeating.
Q. Using Euclid’s Division Algorithm, find the HCF of 867 and 255.
A) 1
B) 3
C) 17
D) 51
Answer: C
Explanation: Applying Euclid’s algorithm repeatedly gives HCF = 17.
Q. If the HCF of 65 and 117 is expressible in the form 65m - 117, then the value of m is:
A) 1
B) 2
C) 3
D) 4
Answer: B
Explanation: HCF of 65 and 117 is 13. Solving 65m - 117 = 13 gives m = 2.
Q. The decimal expansion of the rational number twenty-three divided by (two square × five) will terminate after how many decimal places?
A) One
B) Two
C) Three
D) Four
Answer: B
Explanation: Denominator becomes twenty, so decimal terminates after two places.
Q. If ‘p’ is a prime number, then √p is:
A) Always rational
B) Always irrational
C) Sometimes rational, sometimes irrational
D) An integer
Answer: B
Explanation: Square root of a prime number cannot be expressed as a rational number.
Q. The HCF of 144 and 198 is:
A) 6
B) 9
C) 18
D) 12
Answer: C
Explanation: Prime factorization gives 144 = 2⁴ × 3² and 198 = 2 × 3² × 11. Common factors give HCF = 18.
Q. If ‘p’ and ‘q’ are positive integers, and the prime factorization of ‘p’ is 2² × 3² × 5 and ‘q’ is 2³ × 3⁴ × 7, then what is LCM(p, q)?
A) 2² × 3²
B) 2³ × 3⁴ × 5 × 7
C) 2⁵ × 3⁶ × 5 × 7
D) 2³ × 3⁴
Answer: B
Explanation: LCM takes highest powers of all prime factors present.
Q. A number is called irrational if its decimal expansion is:
A) Terminating
B) Non-terminating recurring
C) Non-terminating non-recurring
D) Always an integer
Answer: C
Explanation: Irrational numbers have decimal expansions that neither terminate nor repeat.
Q. What is the largest number that divides seventy and one hundred twenty-five, leaving remainders five and eight respectively?
A) 13
B) 65
C) 875
D) 1750
Answer: A
Explanation: Required divisor divides 70 - 5 = 65 and 125 - 8 = 117. HCF of 65 and 117 is 13.
Q. Which of the following is not a rational number?
A) √16
B) √12/√3
C) 2 + √4
D) √5
Answer: D
Explanation: √5 is irrational, while all other options simplify to rational numbers.
Q. Express 0.6 as a rational number in simplest form.
A) 6/10
B) 3/5
C) 1/6
D) 6/100
Answer: B
Explanation: 0.6 = 6/10, which simplifies to 3/5.
Key Concepts Students Must Understand Before Solving MCQs
- Euclid’s Division Lemma: According to Euclid’s Division Lemma:
a = bq + r
Where:
“a” is dividend
“b” is divisor
“q” is quotient
“r” is remainder
Condition:
0 ≤ r < b
This concept is mainly used for finding HCF of two numbers and is one of the most important topics for Class 10 Real Numbers important MCQs.
- Fundamental Theorem of Arithmetic: The theorem states that every composite number can be expressed as a product of prime numbers in a unique way.
Example:
540 = 2² × 3³ × 5
Questions based on prime factorization are commonly included in Real Numbers board exam MCQs because they test conceptual understanding instead of direct formulas.
- Terminating and Non-Terminating Decimals: A rational number p/q will have a terminating decimal expansion only if the denominator contains prime factors of:
2
5
or both
Examples:
13/40 → Terminating decimal
7/11 → Non-terminating recurring decimal
Students often confuse recurring decimals with irrational numbers, which leads to wrong answers in exams.
Common Mistakes Students Make in Real Numbers MCQs
Even students who know formulas sometimes lose marks because of small conceptual mistakes. These are some common errors noticed in exams:
- Forgetting the condition:
0 ≤ r < b - Incorrect prime factorization
- Confusing rational and irrational numbers
- Mistakes while identifying terminating decimals
- Writing wrong HCF after using Euclid algorithm
- Ignoring powers of prime numbers in factorization
- Solving too quickly without checking options carefully
One more thing students usually do is memorizing steps instead of understanding the reason behind them. That works in short tests sometimes, but not in competency-based papers.
Preparation Tips for Real Numbers Class 10 MCQs
Students can improve performance in MCQs on Real Numbers Class 10 by following a simple preparation strategy.
- Focus on Concepts First: Do not directly start solving dozens of questions. First understand:
- HCF
- LCM
- Prime factorization
- Decimal expansion
Once the concepts become clear, MCQs feel much easier.
- Practice Mixed Difficulty Questions: Solve:
- Easy MCQs
- Logic-based MCQs
- Competency-based questions
- Assertion reason questions
CBSE is gradually increasing concept application questions, so practicing only direct objective questions is not enough now.
- Revise Important Rules Regularly: Especially revise:
- Euclid Division Lemma
- Conditions for terminating decimals
- Prime factorization rules
These concepts appear again and again in Real Numbers multiple choice questions.
Conclusion
Practicing Real Numbers Class 10 MCQs with answers is one of the best ways to strengthen concepts and improve exam performance. This chapter may seem simple initially, but many questions become tricky when concepts are mixed together. Regular revision, conceptual understanding, and solving different types of objective questions can help students score much better in school exams as well as CBSE board papers. Students preparing seriously for the 2026 exams should focus not only on formulas but also on understanding how and why mathematical concepts work.