CBSE Class 10 Mathematics: Real Numbers — Important Questions with Answers 2026

Short Answer Questions (2-3 Marks)

• Q: State the Fundamental Theorem of Arithmetic.
  Ans: Every composite number can be expressed as a product of primes, and this factorisation is unique apart from the order of the prime factors. Example: 12 = 2² × 3. This is the only way to express 12 as a product of primes. Used for finding HCF and LCM.
• Q: Find the HCF and LCM of 306 and 657 using prime factorisation.
  Ans: 306 = 2 × 3² × 17. 657 = 3 × 219 = 3 × 3 × 73 = 3² × 73. HCF = product of common prime factors with lowest power = 3² = 9. LCM = product of all prime factors with highest power = 2 × 3² × 17 × 73 = 22338. Verification: HCF × LCM = 9 × 22338 = 201042 = 306 × 657 ✓
• Q: Prove that √2 is irrational.
  Ans: Assume √2 is rational. Then √2 = p/q where p, q are co-prime integers and q ≠ 0. Squaring: 2 = p²/q² → p² = 2q². Since p² is even, p must be even. Let p = 2m. Then (2m)² = 2q² → 4m² = 2q² → q² = 2m². So q² is also even, hence q is even. But both p and q being even contradicts our assumption that they are co-prime. Therefore, √2 is irrational.

Exam Tips for This Chapter

• Revise all definitions and laws from Real Numbers — they are commonly asked as 1-2 mark questions
• Practice diagrams related to Real Numbers — neat labelled diagrams carry 2-3 marks
• For numericals, always show formula → substitution → answer with correct units
• Previous year analysis shows Real Numbers carries 8-12 marks in the board exam