Students searching for “NCERT Solutions Class 10 Maths Chapter 1” are almost always stuck on the same handful of questions from Exercise 1.1. Below are complete, step-by-step solutions for the current 2026-27 NCERT Class 10 Maths textbook, Chapter 1 — Real Numbers.
NCERT Solutions for Class 10 Maths Chapter 1: Real Numbers (Exercise 1.1)
Q1. Prove that √5 is irrational.
Assume, for contradiction, that √5 is rational. Then √5 = p/q, where p and q are coprime integers (q ≠ 0). Squaring both sides: 5q² = p², so p² is divisible by 5, which means p itself must be divisible by 5. Let p = 5m. Substituting back: 5q² = 25m², so q² = 5m², meaning q is also divisible by 5. But this contradicts our assumption that p and q are coprime (they now share a common factor of 5). Hence, √5 is irrational.
Q2. Prove that 3 + 2√5 is irrational.
Assume 3 + 2√5 is rational, equal to some rational number r. Then 2√5 = r − 3, so √5 = (r − 3)/2. Since r is rational, (r − 3)/2 is also rational, which would mean √5 is rational — contradicting Q1’s result. Hence, 3 + 2√5 is irrational.
Q3. Given that HCF(306, 657) = 9, find LCM(306, 657).
Using the relation HCF × LCM = product of the two numbers:
LCM = (306 × 657) ÷ 9 = 306 × 73 = 22,338.
Q4. Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.
7 × 11 × 13 + 13 = 13 × (7 × 11 + 1) = 13 × 78 = 1,014. Since it can be written as a product of 13 and 78, it is composite.
7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 = 5 × (7 × 6 × 4 × 3 × 2 × 1 + 1) = 5 × 1,009 = 5,045. Since it can be written as a product of 5 and 1,009, it is composite.
Q5. Check whether 6n can end with the digit 0 for any natural number n.
For a number to end in 0, it must be divisible by 10, i.e. by both 2 and 5. Now 6n = (2 × 3)n = 2n × 3n. The only prime factors of 6n are 2 and 3 — 5 is never a factor, no matter what value n takes. So 6n can never end with the digit 0.
Why This Chapter Matters
Real Numbers is a short but high-yield chapter — it typically contributes 1–2 direct questions in the CBSE Class 10 board Maths paper (proof-of-irrationality and HCF/LCM application questions are recurring favourites), and the proof techniques here (contradiction, prime factorisation) reappear across Class 10 and Class 11 algebra. Students preparing for JEE Foundation-level material also encounter the same number-theory groundwork later.
How to Use These Solutions
Try each question yourself first using a rough sheet, then check your working against the steps above — the goal is to internalise the proof pattern (assume the opposite, reach a contradiction) rather than memorise this specific answer, since board exams frequently rephrase the same question with different numbers. You can download the full chapter directly from our Class 10 Maths NCERT book page, or browse every Class 10 subject on our Class 10 hub.
Want extra practice or a fast recap? See Extra Questions for Real Numbers, our Revision Notes summary, or the Class 10 Maths Formulas Handbook.
Frequently Asked Questions
Are these solutions based on the current 2026-27 NCERT syllabus?
Yes — this covers the rationalised Chapter 1 (Real Numbers) as currently prescribed by NCERT for the 2026-27 academic session, matching the single-exercise structure introduced after the 2023 syllabus rationalisation.
Is Chapter 1 important for the CBSE Class 10 board exam?
Yes. Real Numbers is a compact chapter that consistently appears in board papers, usually as a 2–3 mark proof or HCF/LCM application question, making it high value relative to the small amount of content it covers.
