Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.
CBSE Class 10 Real Numbers — Complete Guide
Real Numbers is the foundation chapter of CBSE Class 10 Mathematics, carrying approximately 6 marks in the board exam. This chapter covers Euclid's division algorithm, the Fundamental Theorem of Arithmetic, irrational number proofs, and decimal expansions of rational numbers.
Quick Revision: Key Concepts
- Euclid's Division Lemma: a = bq + r (0 ≤ r < b) — used to find HCF
- Fundamental Theorem: Every composite number = unique product of primes
- HCF: Common primes with smallest powers | LCM: All primes with greatest powers
- HCF × LCM = Product of two numbers
- Irrationality: √2, √3, √5 are irrational (proof by contradiction)
- Terminating decimals: p/q terminates iff q = 2m × 5n
Board Exam Pattern
Typically 2-3 MCQs (1 mark each) on HCF/LCM and decimal expansions, plus 1 short answer question (2-3 marks) on irrationality proof or Euclid's algorithm. Total: approximately 6 marks.
Most Important Questions
- Find HCF using Euclid's algorithm
- Find HCF and LCM using prime factorisation and verify HCF × LCM = product
- Prove √2 (or √3 or √5) is irrational
- Determine if decimal expansion is terminating or non-terminating
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