CBSE Class 10 Maths Chapter 1 — Real Numbers: NCERT Solutions, Euclids Algorithm and 10 MCQs 2027

Last Updated: May 2026

CBSE Class 10 Maths Chapter 1 — Real Numbers is the gateway chapter of the Class 10 syllabus, and one of the highest-weightage chapters in the Standard paper. The 2026–27 NCERT edition keeps the chapter focused on three pillars: Euclid’s Division Lemma, the Fundamental Theorem of Arithmetic, and Irrational Numbers / Decimal Expansion. This post gives you the complete NCERT solutions framework, the four classes of board questions you must drill, an exam-tested method for each, and 10 MCQs (with answers) at the end. Print this post, work through it once with a pen, and you have a chapter that consistently delivers 6 marks of the 80 in the Standard paper.

Chapter 1 at a Glance

1. Euclid’s Division Lemma — The Foundation

Statement: Given any two positive integers a and b, there exist unique whole numbers q (quotient) and r (remainder) such that a = bq + r, where 0 ≤ r < b.

What this means in plain language: When you divide one positive integer by another, you always get a unique quotient and a unique remainder, and the remainder is always smaller than the divisor.

Euclid’s Division Algorithm — Finding HCF Step by Step

To find the HCF of two positive integers a and b with a > b:

1. Apply the lemma: a = bq₁ + r₁ (with 0 ≤ r₁ < b).
2. If r₁ = 0, then HCF = b.
3. If r₁ ≠ 0, apply the lemma to b and r₁: b = r₁q₂ + r₂.
4. Continue until the remainder becomes 0. The non-zero divisor at that step is the HCF.

Worked Example: HCF of 420 and 272

Step 1: 420 = 272 × 1 + 148
Step 2: 272 = 148 × 1 + 124
Step 3: 148 = 124 × 1 + 24
Step 4: 124 = 24 × 5 + 4
Step 5: 24 = 4 × 6 + 0
The remainder is now 0. The divisor at this step is 4, so HCF(420, 272) = 4.

2. The Fundamental Theorem of Arithmetic

Statement: 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.

This theorem is the most testable concept of the chapter. It powers four of the most-asked board questions.

Use Case A — Finding HCF and LCM by Prime Factorisation

Method: Express both numbers as products of primes; HCF is the product of the smallest power of each common prime; LCM is the product of the highest power of each prime appearing in either number.

Example. Find HCF and LCM of 96 and 404.
96 = 2⁵ × 3; 404 = 2² × 101.
Common prime: 2. Smallest power = 2² = 4. HCF = 4.
LCM = 2⁵ × 3 × 101 = 32 × 303 = 9696.
Verify: HCF × LCM = 4 × 9696 = 38784 = 96 × 404. ✓

Use Case B — Proving Irrationality of √p (where p is prime)

This 2-mark proof appears every year in either Standard or Basic. Memorise the structure exactly:

1. Assume, for contradiction, that √p is rational. Then √p = a/b, where a and b are coprime integers and b ≠ 0.
2. Square both sides: p × b² = a². So p divides a².
3. By the Fundamental Theorem of Arithmetic, since p is prime and p divides a², p must divide a.
4. Write a = p × k for some integer k. Substituting: p × b² = p²k², which gives b² = p × k². So p divides b².
5. Therefore p divides b. But we assumed a and b are coprime — contradiction.
6. Hence √p is irrational.

3. Decimal Expansion of Rational Numbers

Let x = p/q be a rational number in lowest terms. Then:

• x has a terminating decimal expansion if and only if q can be expressed as 2ⁿ × 5ᵐ for some non-negative integers n and m.
• If q has any prime factor other than 2 or 5, x has a non-terminating, repeating decimal expansion.

Example. Is 13/3125 terminating? Factor: 3125 = 5⁵. Only prime is 5, so YES, it terminates. Compute: 13/3125 = 13 × 32 / (3125 × 32) = 416 / 100000 = 0.00416.

Counter-example. 17/30. Factor 30 = 2 × 3 × 5. The prime 3 is present, so the expansion is non-terminating, repeating.

The Four Board-Question Patterns

Word Problem Worked Example

Question. A school has 60 students in Class A, 84 in Class B, and 108 in Class C. The principal wants to seat them in rows where each row has the same number of students and only one class sits in each row. What is the maximum number of students per row?

Solution. “Maximum equal” → HCF.
60 = 2² × 3 × 5; 84 = 2² × 3 × 7; 108 = 2² × 3³.
HCF = 2² × 3 = 12.
Maximum 12 students per row.

10 MCQs — CBSE Class 10 Maths Chapter 1 (2027 Pattern)

1. If HCF (a, b) = 12 and a × b = 1800, then LCM (a, b) is:
   (A) 3600   (B) 900   (C) 150   (D) 90
   Answer: (C) 150 [Use HCF × LCM = a × b ⇒ 12 × LCM = 1800 ⇒ LCM = 150]
2. The decimal expansion of 23 / (2³ × 5²) will terminate after how many decimal places?
   (A) 2   (B) 3   (C) 4   (D) 5
   Answer: (B) 3 [Max of powers of 2 and 5 in denominator = max(3, 2) = 3]
3. The HCF of 96 and 404 is:
   (A) 2   (B) 4   (C) 8   (D) 12
   Answer: (B) 4
4. Which of the following is irrational?
   (A) √81   (B) √25 + √16   (C) 0.142857142857…   (D) √7
   Answer: (D) √7
5. If p is a prime number, then √p is:
   (A) Always rational   (B) Always irrational   (C) Sometimes rational   (D) None
   Answer: (B)
6. The number 6ⁿ, where n is a natural number, can never end with the digit:
   (A) 6   (B) 0   (C) 4   (D) 2
   Answer: (B) 0 [For a number to end in 0, its prime factorisation must include both 2 and 5; 6ⁿ = 2ⁿ × 3ⁿ has no 5]
7. The LCM and HCF of two numbers are 180 and 6. If one number is 30, the other is:
   (A) 36   (B) 18   (C) 12   (D) 24
   Answer: (A) 36 [180 × 6 = 30 × x ⇒ x = 36]
8. The decimal expansion of 17/8 will:
   (A) Terminate after 1 place   (B) Terminate after 3 places   (C) Not terminate   (D) Terminate after 2 places
   Answer: (B) [8 = 2³; max power = 3]
9. If the HCF of 65 and 117 is expressible in the form 65m − 117, the value of m is:
   (A) 1   (B) 2   (C) 3   (D) 4
   Answer: (B) 2 [HCF (65, 117) = 13; 65 × 2 − 117 = 13]
10. The product of two consecutive positive integers is always:
    (A) Odd   (B) Even   (C) Prime   (D) An odd prime
    Answer: (B) Even [One of two consecutives is always even]

Common Mistakes to Avoid

• Forgetting “in lowest terms” when applying the decimal-expansion rule. Always reduce p/q first.
• Assuming HCF × LCM = a + b. The correct identity is HCF × LCM = a × b.
• Skipping “coprime” in the irrationality proof. The contradiction depends on a and b being coprime — drop it and you lose marks.
• Confusing prime factorisation with division. 12 = 2² × 3, not 2 × 6.

Revision Plan: Five Days to Lock Chapter 1

For the full Class 10 Maths chapter map and topic-wise practice tests, head to our free mock test dashboard and the CBSE board exam FAQ.

Frequently Asked Questions

Q1. Is Euclid’s Division Lemma still in the rationalised CBSE Class 10 syllabus?

Yes — the lemma and the algorithm for finding HCF remain in the 2024 rationalised NCERT, although the chapter has been compressed. Focus on its application to HCF rather than abstract derivation.

Q2. Will the proof of irrationality always be on √2, √3 or √5?

Most board papers test √2, √3 or √5 because the NCERT examples use these. The method generalises to any prime, so master the structure rather than memorising one specific case.

Q3. Do I need to remember the formula HCF × LCM = product of two numbers for any numbers?

This identity holds only for two numbers, not three or more. For three numbers, no such simple product identity exists.

Q4. How many marks does this chapter typically carry in the Class 10 Standard paper?

Approximately 6 marks across one short answer, one MCQ, and one long answer (or assertion-reason). It is one of the highest mark-per-effort ratios of the entire syllabus.

Q5. Is Real Numbers easier in Maths Basic compared to Standard?

The conceptual content is the same. Maths Basic tends to use simpler numbers and skip the more abstract proof variants. The irrationality proof and HCF/LCM problems remain.

Master Chapter 1 once, and every subsequent number-theory and algebra chapter becomes easier — Polynomials, Quadratic Equations, even Coordinate Geometry’s section formulas all rely on confident HCF/LCM and prime-factorisation work. Drill the four board patterns, write the irrationality proof from memory, and practise the 10 MCQs above. To structure full Class 10 revision with chapter-wise tests and mentor support, explore our Class 10 board exam courses.