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Practice Class 10 Maths Chapter 1 Real Numbers MCQ Questions with Answers based on NCERT syllabus. Learn HCF, LCM, Euclid Division Lemma, irrational numbers, prime factorization, and decimal expansion with detailed solutions for CBSE and state board exams.
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Chapter Information
Subject: Mathematics
Class: 10
Chapter: Real Numbers
Question Type: Multiple Choice Questions (MCQs)
Difficulty Level: Easy to Moderate
Based On: NCERT Latest Syllabus
Introduction:
The chapter Real Numbers helps students understand the properties of numbers and their applications in mathematics. Concepts such as prime factorization, Euclid’s Division Lemma, HCF, LCM, irrational numbers, and decimal expansions are essential for solving mathematical problems efficiently. These concepts are frequently tested in board examinations and form the basis for many advanced topics. This practice set contains carefully designed MCQs with detailed explanations to strengthen conceptual understanding and improve exam performance.
What You Will Learn?
✔ Euclid’s Division Lemma
✔ Prime Factorization
✔ HCF and LCM
✔ Rational Numbers
✔ Irrational Numbers
✔ Decimal Expansions
✔ Board Exam Oriented Concepts
Why This Topic Is Important?
Real Numbers is a foundational chapter in mathematics. A strong understanding of this chapter helps students solve algebraic problems more effectively and prepares them for higher-level mathematical concepts. Questions from this chapter are commonly asked in school and board examinations.
Exam Relevance
These questions are useful for:
✔ CBSE Board Exams
✔ State Board Exams
✔ School Unit Tests
✔ Half-Yearly Exams
✔ Annual Exams
✔ Scholarship Examinations
Q1. The prime factorization of $$180$$ is:
A) $$2^2 \times 3^2 \times 5$$
B) $$2 \times 3^2 \times 5$$
C) $$2^2 \times 3 \times 5$$
D) $$2^3 \times 3^2 \times 5$$
Answer:
$$2^2 \times 3^2 \times 5$$
Useful Formula for this Question:
Prime factorization means expressing a number as a product of prime numbers only.
Solution:
$$180 = 2 \times 90$$
$$90 = 2 \times 45$$
$$45 = 3 \times 15$$
$$15 = 3 \times 5$$
Therefore:
$$180 = 2 \times 2 \times 3 \times 3 \times 5$$
$$= 2^2 \times 3^2 \times 5$$
Hence, the correct answer is:
$$2^2 \times 3^2 \times 5$$
————————————————–
Q2. Using Euclid’s Division Lemma, when $$53$$ is divided by $$9$$, the remainder is:
A) $$6$$
B) $$7$$
C) $$8$$
D) $$9$$
Answer:
$$8$$
Useful Formula for this Question:
$$a = bq + r$$
where
$$0 \le r < b$$
Solution:
Divide:
$$53 \div 9$$
We get:
$$53 = 9 \times 5 + 8$$
Therefore:
$$r = 8$$
Hence, the correct answer is:
$$8$$
————————————————–
Q3. Which of the following numbers is irrational?
A) $$\frac{8}{15}$$
B) $$0.375$$
C) $$\sqrt{19}$$
D) $$0.818181\ldots$$
Answer:
$$\sqrt{19}$$
Useful Formula for this Question:
An irrational number cannot be written in the form:
$$\frac{p}{q}$$
where $$q \ne 0$$.
Solution:
$$\sqrt{19}$$ cannot be expressed as:
$$\frac{p}{q}$$
Its decimal expansion is non-terminating and non-recurring.
Therefore, it is irrational.
Hence, the correct answer is:
$$\sqrt{19}$$
————————————————–
Q4. The LCM of $$16$$ and $$24$$ is:
A) $$32$$
B) $$48$$
C) $$64$$
D) $$96$$
Answer:
$$48$$
Useful Formula for this Question:
LCM is obtained by taking the highest powers of all prime factors.
Solution:
Prime factorization:
$$16 = 2^4$$
$$24 = 2^3 \times 3$$
Taking highest powers:
$$LCM = 2^4 \times 3$$
$$= 16 \times 3$$
$$= 48$$
Therefore, the correct answer is:
$$48$$
————————————————–
Q5. Which of the following fractions has a terminating decimal expansion?
A) $$\frac{7}{45}$$
B) $$\frac{13}{50}$$
C) $$\frac{11}{18}$$
D) $$\frac{17}{27}$$
Answer:
$$\frac{13}{50}$$
Useful Formula for this Question:
A rational number has a terminating decimal expansion if the denominator contains only the prime factors:
$$2$$ and/or $$5$$
Solution:
For:
$$\frac{13}{50}$$
Denominator:
$$50 = 2 \times 5^2$$
Only factors $$2$$ and $$5$$ are present.
Therefore, the decimal expansion terminates.
$$\frac{13}{50}=0.26$$
Hence, the correct answer is:
$$\frac{13}{50}$$
————————————————–
Important Formulas and Concepts
Euclid’s Division Lemma:
$$a = bq + r$$
where
$$0 \le r < b$$
Fundamental Theorem of Arithmetic:
Every composite number can be expressed as a unique product of prime numbers.
Relationship between HCF and LCM:
$$HCF \times LCM = Product\ of\ two\ numbers$$
A rational number is of the form:
$$\frac{p}{q}$$
where $$q \ne 0$$
————————————————–
FAQs
Q. What is the Fundamental Theorem of Arithmetic?
Answer:
Every composite number can be expressed as a unique product of prime numbers.
Q. What is an irrational number?
Answer:
A number that cannot be expressed in the form:
$$\frac{p}{q}$$
Q. How do we find the LCM?
Answer:
Take the highest powers of all prime factors appearing in the given numbers.
————————————————–
Common Mistakes Students Make
❌ Confusing HCF with LCM.
❌ Ignoring repeated prime factors.
❌ Writing a remainder equal to the divisor.
❌ Assuming every decimal is terminating.
————————————————–
Quick Revision Notes
✔ Prime factorization uses only prime numbers.
✔ Euclid’s Division Lemma:
$$a = bq + r$$
✔ Remainder is always less than the divisor.
✔ Irrational numbers cannot be written as:
$$\frac{p}{q}$$
✔ Terminating decimals have denominators with only factors $$2$$ and/or $$5$$.
—————————————–
Conclusion:
These Class 10 Maths Chapter 1 Real Numbers MCQs with detailed solutions help students develop a strong understanding of prime factorization, Euclid’s Division Lemma, irrational numbers, HCF, LCM, and decimal expansions. Regular practice can improve accuracy, confidence, and performance in board examinations.
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