Class 10 Maths Chapter 1, Real Numbers, opens the NCERT Class 10 Mathematics textbook (2026-27) with the Fundamental Theorem of Arithmetic, prime factorisation for finding HCF and LCM, and proofs by contradiction that √2, √3 and √5 are irrational.
What Class 10 Mathematics (Ganit — Hindi Medium) Chapter 1 Covers
1.1 & 1.2 The Fundamental Theorem of Arithmetic states that every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique apart from the order in which the prime factors occur. It uses prime factorisation to compute HCF(a,b) and LCM(a,b), and proves the key relation HCF(a,b) × LCM(a,b) = a × b for any two positive integers a and b (and checks why 4^n or 6^n can never end with the digit 0).
1.3 Revisiting Irrational Numbers uses Theorem 1.2 (if a prime p divides a2, then p divides a) to prove by contradiction that √2, √3, √5, 5 – √3 and 3√2 are irrational numbers.
Exercises in Chapter 1
- Exercise 1.1 (7 Questions) covers prime factorisation trees, verifying HCF × LCM = product of the two numbers, proving 7× 11× 13 + 13 is composite, and circular-path meeting time (LCM(18, 12) = 36 minutes).
- Exercise 1.2 (3 Questions) proves the irrationality of √5, 3 + 2√5, 1/√2, 7√5 and 6 + √2.
Read Class 10 Mathematics (Ganit — Hindi Medium) Chapter 1 Online
More Chapters
- Next: Chapter 2: बहुपद (Polynomials)
- English medium: Class 10 Maths Chapter 1 PDF in English
- All Mathematics (Ganit — Hindi Medium) chapters and the other Class 10 books: Class 10 NCERT books 2026-27
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FAQ
What is the relationship between HCF and LCM of two positive integers?
For any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b. Note that this formula applies directly to two numbers, not three.
Why can 6^n never end with the digit 0 for any natural number n?
A number ending with the digit 0 must be divisible by 10 = 2 × 5, so its prime factorisation must contain both 2 and 5. Since 6^n = (2 × 3)^n = 2^n × 3^n has only 2 and 3 as prime factors, by uniqueness of the Fundamental Theorem of Arithmetic it can never end with 0.