Class 10 Maths Chapter 1 Real Numbers, HCF, LCM,

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Practice Class 10 Maths Chapter 1 Real Numbers MCQ Questions with Answers based on NCERT syllabus. Learn HCF, LCM, Euclid’s Division Lemma, prime factorization, irrational numbers, and decimal expansion with concept-based solutions and exam tips for CBSE board exams.

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Chapter Information

Subject: Mathematics

Class: 10

Chapter: Real Numbers

Question Type: Multiple Choice Questions (MCQs)

Difficulty Level: Moderate

Based On: NCERT Latest Syllabus

Introduction:

Real Numbers is one of the most fundamental chapters in Class 10 Mathematics. It helps students understand the properties of numbers, prime factorization, Euclid’s Division Lemma, HCF, LCM, irrational numbers, and decimal expansions. Questions from this chapter are frequently asked in school examinations and board exams. A strong understanding of these concepts helps students perform better in mathematics and builds a foundation for higher-level topics.

What You Will Learn?

✔ Prime Factorization

✔ Euclid’s Division Lemma

✔ HCF and LCM

✔ Rational Numbers

✔ Irrational Numbers

✔ Decimal Expansion

✔ Board Exam Preparation

Why This Topic Is Important?

The concepts learned in this chapter are used throughout mathematics. Understanding Real Numbers improves logical thinking and helps students solve algebraic and arithmetic problems more efficiently.

Exam Relevance

These questions are useful for:

✔ CBSE Board Exams

✔ State Board Exams

✔ School Tests

✔ Half-Yearly Exams

✔ Annual Exams

✔ Scholarship Examinations

Q1. The prime factorization of $$210$$ is:

A) $$2 \times 3 \times 5 \times 7$$

B) $$2^2 \times 3 \times 5 \times 7$$

C) $$2 \times 3^2 \times 5 \times 7$$

D) $$2 \times 5 \times 7$$

Answer:

$$2 \times 3 \times 5 \times 7$$

Useful Formula for this Question:

Prime factorization means expressing a number as a product of prime numbers only.

Concept Behind This Question:

This question checks whether students can break a composite number into its prime factors. Prime factorization is an important skill for finding HCF and LCM.

Solution:

Step 1:

$$210 = 2 \times 105$$

Step 2:

$$105 = 3 \times 35$$

Step 3:

$$35 = 5 \times 7$$

Therefore:

$$210 = 2 \times 3 \times 5 \times 7$$

Hence, the correct answer is:

$$2 \times 3 \times 5 \times 7$$

Exam Tip:

Always continue factorization until every factor is a prime number.

Q2. The HCF of $$56$$ and $$98$$ is:

A) $$7$$

B) $$14$$

C) $$21$$

D) $$28$$

Answer:

$$14$$

Useful Formula for this Question:

HCF is obtained by taking common prime factors with the smallest powers.

Concept Behind This Question:

Students must identify common prime factors and select their lowest powers while finding the HCF.

Solution:

Prime factorization:

$$56 = 2^3 \times 7$$

$$98 = 2 \times 7^2$$

Common prime factors:

$$2 \times 7$$

$$= 14$$

Therefore:

$$HCF = 14$$

Exam Tip:

For HCF, always take the lowest power of common prime factors.

Q3. Which of the following numbers is irrational?

A) $$\frac{5}{8}$$

B) $$0.75$$

C) $$\sqrt{23}$$

D) $$0.4444\ldots$$

Answer:

$$\sqrt{23}$$

Useful Formula for this Question:

An irrational number cannot be written in the form:

$$\frac{p}{q}$$

Concept Behind This Question:

Students should be able to distinguish between rational and irrational numbers.

Solution:

$$\sqrt{23}$$ is the square root of a non-perfect square.

Therefore, it cannot be written as:

$$\frac{p}{q}$$

Hence, it is irrational.

Therefore, the correct answer is:

$$\sqrt{23}$$

Exam Tip:

The square root of a non-perfect square is usually irrational.

Q4. The LCM of $$18$$ and $$30$$ is:

A) $$60$$

B) $$90$$

C) $$120$$

D) $$180$$

Answer:

$$90$$

Useful Formula for this Question:

LCM is obtained using the highest powers of all prime factors.

Concept Behind This Question:

Students must understand how prime factorization is used to find the least common multiple.

Solution:

$$18 = 2 \times 3^2$$

$$30 = 2 \times 3 \times 5$$

Taking highest powers:

$$LCM = 2 \times 3^2 \times 5$$

$$= 90$$

Therefore, the correct answer is:

$$90$$

Exam Tip:

For LCM, always take the highest power of every prime factor present.

Q5. Which of the following fractions has a terminating decimal expansion?

A) $$\frac{7}{24}$$

B) $$\frac{9}{50}$$

C) $$\frac{11}{42}$$

D) $$\frac{13}{27}$$

Answer:

$$\frac{9}{50}$$

Useful Formula for this Question:

A rational number has a terminating decimal expansion if the denominator contains only:

$$2$$ and/or $$5$$

Concept Behind This Question:

Students should know how denominator factorization affects decimal expansion.

Solution:

$$50 = 2 \times 5^2$$

Only factors $$2$$ and $$5$$ are present.

Therefore, the decimal expansion terminates.

Hence, the correct answer is:

$$\frac{9}{50}$$

Exam Tip:

Check the denominator after simplification before deciding the type of decimal expansion.

(Important Formulas, FAQs, Common Mistakes, Quick Revision Notes, and Conclusion same structure as previous sets.)

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