NCERT Solutions Class 10 Maths Chapter 1 provides detailed solutions to all 10 questions across 2 exercises on Real Numbers. You’ll learn how to apply Euclid’s Division Lemma to find HCF of large numbers, prove the Fundamental Theorem of Arithmetic, and determine whether rational numbers have terminating or non-terminating decimal expansions. Each solution includes the exact steps used in CBSE exams, plus shortcuts for solving HCF and LCM problems faster.
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All exercises with step-by-step solutions | Updated 2025-26 | Free Download
Download PDF (Free)NCERT Solutions Class 10 Maths Chapter 1 Real Numbers – Complete Guide
NCERT Class 10 Chapter 1 Real Numbers forms the foundation of your CBSE board mathematics journey. You’ll explore the fascinating world of numbers, starting with Euclid’s Division Lemma and its applications in finding the Highest Common Factor (HCF) of two positive integers. This chapter carries 6 marks in the CBSE board exam and is considered of high importance due to its fundamental concepts that appear throughout your Class 10 syllabus.
📊 CBSE Class 10 Maths Chapter 1 – Exam Weightage & Marking Scheme
| CBSE Board Marks | 6 Marks |
| Unit Name | Number Systems |
| Difficulty Level | Medium |
| Importance | High |
| Exam Types | CBSE Board, State Boards |
| Typical Questions | 2-3 questions |
You will dive deep into the Fundamental Theorem of Arithmetic, which states that every composite number can be expressed as a product of prime numbers in a unique way. This powerful theorem helps you understand the structure of numbers and solve complex problems involving HCF and LCM. You’ll also learn to distinguish between rational and irrational numbers by examining their decimal expansions—terminating, non-terminating repeating, and non-terminating non-repeating decimals.
The chapter includes various question types in CBSE exams: 2-mark questions on Euclid’s division algorithm, 3-mark questions on proving irrationality, and case-study based MCQs worth 4 marks. You’ll encounter practical applications like finding the least number divisible by given numbers, determining when a rational number has a terminating decimal expansion, and solving real-world problems using HCF and LCM concepts.
Quick Facts – Class 10 Chapter 1
| 📖 Chapter Number | Chapter 1 |
| 📚 Chapter Name | Real Numbers |
| ✏️ Total Exercises | 2 Exercises |
| ❓ Total Questions | 10 Questions |
| 📅 Updated For | CBSE Session 2025-26 |
Mastering Real Numbers is crucial as these concepts connect directly to Polynomials (Chapter 2) and appear in mensuration and algebra problems throughout the year. With consistent practice of NCERT solutions and understanding the logical reasoning behind each theorem, you’ll build strong analytical skills that will help you score full marks in this high-weightage chapter and excel in your CBSE Class 10 board examination.
NCERT Solutions Class 10 Maths Chapter 1 – All Exercises PDF Download
Download exercise-wise NCERT Solutions PDFs for offline study
| Exercise No. | Topics Covered | Download PDF |
|---|---|---|
| EXERCISE 1.1 | Complete step-by-step solutions for 7 questions | 📥 Download PDF |
| EXERCISE 1.2 | Complete step-by-step solutions for 3 questions | 📥 Download PDF |
Real Numbers – Key Formulas & Concepts
Quick reference for CBSE exams
| Formula | Description | When to Use |
|---|---|---|
| Euclid’s Division Lemma \(a = bq + r, \text{ where } 0 \leq r < b\) | Expresses a number ‘a’ in terms of another number ‘b’, quotient ‘q’ and remainder ‘r’. Note: ‘a’ is the dividend, ‘b’ is the divisor, ‘q’ is the quotient, and ‘r’ is the remainder. Remember that the remainder ‘r’ is always non-negative and less than the divisor ‘b’. | To find the HCF (Highest Common Factor) of two numbers or to prove divisibility properties. |
| HCF using Euclid’s Algorithm \(\text{HCF}(a, b) = \text{HCF}(b, r)\) | Iteratively applies Euclid’s Division Lemma to find the HCF of two numbers. Note: Keep applying the lemma until the remainder becomes zero. The divisor at that stage is the HCF. | Specifically to calculate HCF of two given numbers using Euclid’s algorithm. |
| Fundamental Theorem of Arithmetic \(n = p_1^{a_1} \cdot p_2^{a_2} \cdot … \cdot p_k^{a_k}\) | Every composite number can be expressed as a product of prime numbers, and this factorization is unique. Note: \(p_1, p_2, …, p_k\) are distinct prime numbers and \(a_1, a_2, …, a_k\) are their respective exponents. | To find the prime factorization of a composite number, to find HCF and LCM, to prove irrationality. |
| HCF and LCM relation \(\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b\) | The product of the HCF and LCM of two numbers is equal to the product of the numbers. Note: Only applicable for two numbers. Don’t use for three or more numbers directly. | To find the LCM if HCF is known, or vice versa, given the two numbers. |
| LCM Calculation \(\text{LCM}(a, b) = \frac{a \times b}{\text{HCF}(a, b)}\) | Calculates the Least Common Multiple (LCM) of two numbers using their HCF. Note: Make sure you have already calculated or are given the HCF. | When you know the HCF of two numbers and need to find their LCM quickly. |
| Proof of Irrationality (√p) Assume \(\sqrt{p}\) is rational, then derive a contradiction. | Method to prove that numbers like \(\sqrt{2}, \sqrt{3}, \sqrt{5}\) etc. are irrational. Note: The proof involves assuming the opposite (that it’s rational), expressing it as a fraction in simplest form, and showing that this leads to a contradiction. | When the question specifically asks you to prove that a given square root of a prime number is irrational. |
| Proof of Irrationality (a + b√p) Assume \(a + b\sqrt{p}\) is rational, then derive a contradiction. | Method to prove that numbers of the form \(a + b\sqrt{p}\) are irrational (where ‘a’ and ‘b’ are rational and \(\sqrt{p}\) is irrational). Note: Similar to the previous proof, assume it’s rational, isolate the \(\sqrt{p}\) term, and show that this forces \(\sqrt{p}\) to be rational (which is a contradiction). | When the question asks you to prove the irrationality of a number in the form \(a + b\sqrt{p}\) such as \(2 + \sqrt{3}\) or \(5 – 2\sqrt{2}\). |
| Decimal Expansion: Terminating A rational number \(\frac{p}{q}\) has a terminating decimal expansion if \(q = 2^m 5^n\), where m and n are non-negative integers. | Condition for a rational number to have a terminating decimal expansion. Note: The denominator ‘q’ must be expressible in the form \(2^m 5^n\). If it has any other prime factor, the decimal expansion will be non-terminating repeating. | To determine whether a given rational number will have a terminating decimal expansion WITHOUT performing actual division. |
| Decimal Expansion: Non-Terminating Repeating A rational number \(\frac{p}{q}\) has a non-terminating repeating decimal expansion if \(q\) has prime factors other than 2 and 5. | Condition for a rational number to have a non-terminating repeating decimal expansion. Note: If the denominator ‘q’ has prime factors other than 2 and 5, the decimal expansion will be non-terminating repeating. Make sure the fraction p/q is in its simplest form. | To determine whether a given rational number will have a non-terminating repeating decimal expansion WITHOUT performing actual division. |
Frequently Asked Questions – NCERT Class 10 Maths Chapter 1
📚 Related Study Materials – Class 10 Maths Resources
| Resource | Access |
|---|---|
| NCERT Class 10 Maths All Chapters | View Solutions |
| NCERT Class 10 Science Solutions | View Solutions |
| NCERT Class 10 Social Science | View Solutions |
| NCERT Class 10 English Solutions | View Solutions |