Class 10th math chapter 1 Revision Notes

CH – 1 REAL NUMBERS

Class 10 Mathematics — CBSE Board Revision

“Master the concepts. Practice smarter. Revise with confidence.”

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Chapter Overview

In this chapter, you will learn:

• What real numbers are and how they are classified

• The Fundamental Theorem of Arithmetic

• Prime factorisation and finding HCF and LCM

• The relationship between HCF and LCM

• Rational vs. irrational numbers and irrationality proofs

🎯 BOARD EXAM FOCUS

• HCF and LCM via prime factorisation

• HCF(a, b) × LCM(a, b) = a × b

• Proving numbers like √2 or 5 – √3 are irrational

• Word problems on HCF and LCM

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Fundamental Theorem of Arithmetic

Every composite number can be uniquely expressed as a product of prime numbers, regardless of order.

Step-by-Step:

1. Divide by the smallest prime factor (2, 3, 5, etc.).
2. Continue until the quotient is 1.
3. Write the product using exponents.

Example: 140 = 2 × 2 × 5 × 7 = 2² × 5 × 7

Board Exam Tip: Always use exponential form (e.g., 2² × 5 × 7) to avoid errors in HCF/LCM.

⭐ REMEMBER Prime factorisation of any composite number is completely unique.

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HCF & LCM Using Prime Factorisation

• HCF: Product of the smallest power of each common prime factor.

• LCM: Product of the greatest power of each prime factor present.

Example: Find HCF and LCM of 36 and 84. • 36 = 2² × 3² • 84 = 2² × 3¹ × 7¹

HCF(36, 84) = 2² × 3¹ = 12 LCM(36, 84) = 2² × 3² × 7¹ = 252

⚠️ COMMON MISTAKE Including non-common prime factors in HCF. Only include factors present in ALL numbers.

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Important Formula

For any two positive integers a and b: HCF(a, b) × LCM(a, b) = a × b

Example: If HCF = 16 and product = 3072: 16 × LCM = 3072 ⟹ LCM = 192

⚠️ Note: This formula applies ONLY to two numbers, never to three.

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Rational & Irrational Numbers

• Rational Numbers: Can be written as a/b (b ≠ 0). Examples: 3/4, -7, 5.

• Irrational Numbers: Cannot be written as a/b. Examples: √2, √3, √5, π.

Key Rules:

• Rational ± Irrational = Irrational (e.g., 5 – √3)

• Non-zero Rational × Irrational = Irrational (e.g., 3√2)

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Proof of Irrationality

Example: Prove √2 is irrational.

1. Assume √2 = a/b, where a and b are coprime integers (b ≠ 0).
2. Square both sides: a² = 2b² ⟹ 2 divides a² ⟹ 2 divides a.
3. Let a = 2c. Then (2c)² = 2b² ⟹ 2c² = b² ⟹ 2 divides b.
4. Both a and b have 2 as a common factor, contradicting that they are coprime.
5. Thus, √2 is irrational.

🧠 EASY WAY TO REMEMBER

1. State opposite (a/b)
2. Square both sides
3. Show shared factor
4. Stop & contradict

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🔥 Most Important Topics for Boards

⭐⭐⭐ VERY IMPORTANT

• Irrationality proofs (√2, √3, √5, 3 + 2√5)

• Finding HCF/LCM and applying HCF × LCM = a × b

⭐⭐ IMPORTANT

• Expressing numbers as products of primes

• Word problems (LCM for repeating cycles, HCF for maximum grouping)

⭐ IMPORTANT

• Checking if 6ⁿ or 4ⁿ can end in zero (requires factors 2 and 5)

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⚠️ Common Mistakes Students Make

1. Applying HCF × LCM = a × b × c to three numbers (Only valid for two).
2. Confusing powers (HCF uses smallest common power; LCM uses largest of all).
3. Forgetting to state “a and b are coprime integers” in proofs.
4. Mixing up word problems (Use HCF for “largest/maximum”, LCM for “minimum time/repeat”).
5. Vague answers for 6ⁿ ending in 0 (Must specify missing factor 5).

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⚡ Real Numbers Quick Revision Sheet

• Prime Factorisation: Composite = Unique product of primes

• Formula: HCF(a, b) × LCM(a, b) = a × b

• HCF: Smallest power of common prime factors

• LCM: Greatest power of all prime factors

• Theorem 1.2: If prime p divides a², then p divides a

• Coprime: HCF = 1

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📝 Test Yourself

1. Express 3825 as a product of prime factors.
2. Given HCF(96, 404) = 4, find LCM(96, 404).
3. Can 4ⁿ end with 0 for any natural number n? Explain.
4. Can two numbers have HCF = 15 and LCM = 175?
5. Explain why 7 × 11 × 13 + 13 is composite.
6. Prove √5 is irrational.
7. Prove 3 + 2√5 is irrational.
8. Find the smallest number divisible by both 306 and 657.
9. Three bells ring at intervals of 9, 12, and 15 mins. When will they ring together next?
10. Find max stack capacity for 420 kaju burfis and 130 badam burfis.

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🎯 Before Your Exam

Checklist:

☐ I can perform prime factorisation.

☐ I can calculate HCF and LCM.

☐ I know how to use HCF × LCM = a × b.

☐ I can write the full proof of irrationality.

☐ I can solve HCF/LCM word problems.

Believe in your preparation and write clearly. Good luck!