Real Numbers – Class 10 Mathematics

Real Numbers — Class 10 Mathematics

YTC Education study lesson aligned to the current NCERT Class 10 Mathematics chapter and CBSE board competencies.

1. What you must learn

  • Prime factorisation and the Fundamental Theorem of Arithmetic
  • Using prime factors to reason about HCF and LCM
  • Proofs of irrationality, especially √2, √3 and √5
  • Board-style reasoning with real numbers

2. Quick Concept Notes

Fundamental Theorem of Arithmetic: every composite number can be expressed as a product of primes, and apart from the order of factors this representation is unique.

Irrational number: a real number that cannot be written as p/q where p and q are integers and q ≠ 0. Standard proof questions often use contradiction.

3. YTC Worked Practice

Question 1

Find the HCF and LCM of 72 and 120 using prime factorisation.

Solution: 72 = 2³ × 3² and 120 = 2³ × 3 × 5. HCF = 2³ × 3 = 24. LCM = 2³ × 3² × 5 = 360. Check: 24 × 360 = 72 × 120.

Question 2

Explain why √3 is irrational.

Solution idea: Assume √3 = p/q in lowest terms. Squaring gives p² = 3q², so 3 divides p. Write p = 3k. Substitution then shows 3 also divides q, contradicting that p/q was in lowest terms. Therefore √3 is irrational.

Question 3

Can 6ⁿ end in 0 for a positive integer n?

Solution: A number ending in 0 must contain both 2 and 5 as prime factors. But 6ⁿ = 2ⁿ × 3ⁿ has no factor 5. Hence it cannot end in 0.

4. Board Practice

  1. Find HCF and LCM of 96 and 404 by prime factorisation.
  2. Prove that √5 is irrational.
  3. Show that 3 + 2√5 is irrational.
  4. Use prime factorisation to determine the greatest number that divides 245 and 1029 leaving the same remainder.
  5. Explain why the product of two irrational numbers need not always be irrational.

5. Answers / Hints

Q1: 96 = 2⁵×3 and 404 = 2²×101; HCF = 4 and LCM = 9696. Q2–Q3: use contradiction. Q4: subtract the two numbers first and analyse the required divisor. Q5: use an example such as √2 × √2.

6. PYQ Policy

YTC labels a question as an actual CBSE PYQ only after its year and source paper are verified. The questions above are original board-style practice and are not presented as past-paper questions.

Reference alignment: NCERT Mathematics Class 10, Chapter 1 Real Numbers; CBSE Mathematics curriculum.

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