Real Numbers Class 10 Notes (2026-27) — CBSE
Class 10 Maths Chapter 1 notes: the Fundamental Theorem of Arithmetic, HCF and LCM by prime factorisation, and proving numbers like root 2 are irrational.
Real Numbers — Class 10 Maths Notes
Chapter Snapshot
This chapter deepens your understanding of integers and real numbers using the Fundamental Theorem of Arithmetic (unique prime factorisation). You use it to find HCF and LCM and to prove that numbers like √2 are irrational.
Board relevance: part of the Number Systems unit (~6 marks). The two dependable questions are an HCF/LCM problem and an "prove that √n is irrational" proof. Both are fully scoring if you follow the standard steps.
Syllabus note (rationalised): Euclid's division lemma and algorithm and rational numbers and their decimal expansions (terminating / non-terminating recurring) have been removed. Focus on the Fundamental Theorem of Arithmetic and irrationality proofs.
Key Concepts & Definitions
Natural numbers, integers, rational and irrational numbers together make up the real numbers.
- A rational number can be written as p/q where p, q are integers and q ≠ 0.
- An irrational number cannot be written as p/q (e.g. √2, √3, π). Its decimal is non-terminating and non-repeating.
Prime number — a number greater than 1 with exactly two factors, 1 and itself. Composite number — has more than two factors.
The Fundamental Theorem of Arithmetic — every composite number can be expressed as a product of primes, and this factorisation is unique except for the order of the factors.
3825 = 3 × 3 × 5 × 5 × 17 = 3² × 5² × 17
This uniqueness is what makes prime factorisation a reliable tool for HCF and LCM.
Formulas
HCF and LCM by prime factorisation
- HCF (Highest Common Factor) = product of the smallest power of each common prime factor.
- LCM (Lowest Common Multiple) = product of the greatest power of each prime factor involved.
The key relation (two numbers only)
HCF(a, b) × LCM(a, b) = a × b
- Use it to find the LCM when the HCF and the two numbers are known: LCM = (a × b) / HCF.
- ⚠️ This relation holds for two numbers, not for three or more.
Worked Examples
Example 1 — HCF and LCM of 96 and 404:
96 = 2⁵ × 3; 404 = 2² × 101.
- HCF = 2² = 4 (smallest power of the only common prime, 2).
- LCM = (96 × 404) / 4 = 9696.
Example 2 — HCF and LCM of 6, 72, 120 (three numbers):
6 = 2 × 3; 72 = 2³ × 3²; 120 = 2³ × 3 × 5.
- HCF = 2¹ × 3¹ = 6 (smallest powers of common primes 2 and 3).
- LCM = 2³ × 3² × 5 = 360 (greatest power of every prime).
Example 3 — Prove √2 is irrational (by contradiction):
Assume √2 is rational: √2 = a/b, where a, b are integers with no common factor (b ≠ 0).
Then 2 = a²/b² → a² = 2b², so a² is even → a is even. Write a = 2c.
Then (2c)² = 2b² → 4c² = 2b² → b² = 2c², so b² is even → b is even.
But then a and b are both even — they share the factor 2 — contradicting "no common factor". So our assumption is false, and √2 is irrational. ∎
The same argument proves √3, √5, and √p (for any prime p) irrational. Also, a rational + irrational is irrational (e.g. 5 − √3), and a non-zero rational × irrational is irrational (e.g. 3√2).
Example 4 — Word problem (HCF): Two tankers contain 850 litres and 680 litres of diesel. Find the maximum capacity of a container that can measure the diesel of both tankers an exact number of times.
This asks for the largest number that divides both → the HCF. 850 = 2 × 5² × 17, 680 = 2³ × 5 × 17. Common primes with smallest powers: 2¹ × 5¹ × 17 = 170 litres.
Example 5 — Word problem (LCM): In a sports meet, three athletes step with strides of 40 cm, 42 cm, and 45 cm. What is the minimum distance each should cover so that all can cover it in complete steps?
This asks for the least common length → the LCM. 40 = 2³ × 5, 42 = 2 × 3 × 7, 45 = 3² × 5. LCM = 2³ × 3² × 5 × 7 = 2520 cm. So each athlete covers 2520 cm in a whole number of strides.
These two examples show the core skill: "largest that divides / greatest measure" is always HCF, while "smallest common / least that is a multiple" is always LCM.
Important Question Patterns
1. HCF/LCM (2–3 marks): find HCF and LCM by prime factorisation; use HCF × LCM = a × b to find the missing quantity.
2. Word problem (3 marks): "largest number that divides…" → HCF; "smallest number/least time when events coincide…" → LCM (e.g. bells tolling together, runners meeting at the start).
3. Prove irrationality (3 marks): √2, √3, √5, or "prove 5 − √3 is irrational" and "prove 3√2 is irrational" — always start "Let us assume, to the contrary, that … is rational".
4. Fundamental Theorem application (2 marks): show a number like 6ⁿ can never end in 0 (it has no factor 5), or check whether a given number can end in a particular digit.
5. Prime factorisation (1–2 marks): express a composite number as a product of primes.
⚡ Quick Revision
- Fundamental Theorem of Arithmetic: every composite number = a unique product of primes.
- HCF = product of smallest powers of common primes; LCM = product of greatest powers of all primes.
- HCF × LCM = product of the two numbers — for two numbers only.
- Word problems: "largest number that divides / greatest measure" → HCF; "least number / smallest common time" → LCM.
- Irrationality proof (by contradiction): assume = a/b in lowest terms → derive that a and b share a factor → contradiction. Works for √2, √3, √5, √p.
- rational ± irrational = irrational; non-zero rational × irrational = irrational.
- Removed from syllabus: Euclid's division lemma/algorithm and decimal-expansion classification.
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