Textbook chapter 1
Discover how prime factors describe whole numbers and help solve grouping and repetition problems. Then use a contradiction argument to explain why some square roots cannot be fractions.
Textbook exercise solutions
- Exercise 1.1 — 7 question groups
- Exercise 1.2 — 3 question groups
Videos, notes and tests
English · Session 2026–27 · NCERT-aligned study guide
Chapter overview
Discover how prime factors describe whole numbers and help solve grouping and repetition problems. Then use a contradiction argument to explain why some square roots cannot be fractions.
Before you begin: Factors, multiples, prime numbers, fractions and square roots.
Key concepts · Solved examples · Practice · Mind map · Timed chapter test
Learn the key concepts
Prime factorisation
Break a composite integer into prime factors until no composite factor remains. The prime factors, including their repetitions, are unique apart from their order. For example, 180 = 2² × 3² × 5. The number 1 is neither prime nor composite.
HCF and LCM
In prime-factor form, use the smallest common exponents for the HCF and the largest exponents present for the LCM. HCF answers equal-grouping questions with no leftovers; LCM answers when repeating events first coincide.
Irrationality by contradiction
To prove √p irrational for a prime p, assume √p = a/b with coprime integers a,b and b ≠ 0. Squaring gives a² = pb². Thus p divides a, so write a = pk. Substitution implies p divides b too, contradicting coprimality.
Formula and result checklist
- For positive integers a,b: HCF(a,b) × LCM(a,b) = ab.
- If p is prime and p divides a², then p divides a.
Solved examples
Example 1
Find the HCF and LCM of 84 and 126.
84 = 2² × 3 × 7; 126 = 2 × 3² × 7. HCF = 2 × 3 × 7 = 42. LCM = 2² × 3² × 7 = 252. Check: 42 × 252 = 84 × 126 = 10584.
Example 2
Two bells ring every 12 and 18 minutes. They ring together at 9:00. Find the next common ringing time.
LCM(12,18) = 2² × 3² = 36 minutes. The next common ringing is at 9:36.
Example 3
Explain why 3 + 2√5 is irrational.
Assume 3 + 2√5 = r for a rational r. Then √5 = (r − 3)/2 would be rational. This contradicts the irrationality of √5, so the original number is irrational.
Try it yourself
Work on paper before opening each answer. Show the reasoning and include units when needed.
1. Find HCF(72,120).
Check answer 1
24; 72 = 2³ × 3² and 120 = 2³ × 3 × 5.
2. Find LCM(15,24).
Check answer 2
120.
3. Prove √3 irrational.
Check answer 3
Assume √3 = a/b in lowest terms. Then a² = 3b², so 3 divides a. Writing a = 3k gives b² = 3k², so 3 divides b too: a contradiction.
Common mistakes to avoid
- Do not use the two-number product rule unchanged for three numbers.
- An irrational number need not have an irrational square.
Chapter mind map
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Take the 6-question, 10-minute chapter practice test. Each correct answer earns 1 mark; there is no negative marking. See explanations after submission.
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Read the corresponding chapter in the official NCERT textbook, then complete its exercises on paper. These NSL explanations and original practice questions support your textbook study; they are not an exercise-by-exercise solution manual.
Official CBSE syllabus 2026–27 · All chapters · Next chapter
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