Worked example 1
Question: Find the HCF and LCM of 84 and 126 using prime factorisation.
Solution: 84 = 2² × 3 × 7; 126 = 2 × 3² × 7. HCF = 2 × 3 × 7 = 42. LCM = 2² × 3² × 7 = 252. Check: 42 × 252 = 84 × 126.
What this chapter covers
Study prime factorisation, the Fundamental Theorem of Arithmetic, and standard proofs that selected square roots are irrational.
Fundamental Theorem of Arithmetic
Every composite positive integer can be written as a product of primes, and this factorisation is unique apart from the order of the factors. Use prime powers to compare HCF and LCM: the HCF takes the smaller exponent of each shared prime; the LCM takes the larger exponent.
Proving a square root irrational
Assume √p = a/b in lowest terms, where p is prime. Squaring gives a² = p b². Prime factorisation then forces p to divide a, and substitution forces p to divide b, contradicting that a/b is in lowest terms. This method applies to √2, √3 and √5.
Key formulae: For positive integers a,b: HCF(a,b) × LCM(a,b) = a × b. For coprime a,b, HCF = 1.
Worked examples: CBSE question patterns
These are original practice questions modeled on common board-exam patterns; they are not reproduced past-paper questions.
Worked example 2
Question: Show that √5 is irrational.
Solution: Suppose √5 = a/b in lowest terms. Then a² = 5b², so 5 divides a; put a = 5k. Then 25k² = 5b², giving b² = 5k², so 5 divides b too. This contradicts gcd(a,b)=1; therefore √5 is irrational.
Practical activity / investigation
Use 24 counters. Arrange them into equal rows in every possible way and write each factor pair. Repeat with 36. Compare the prime-power forms and use them to predict the HCF and LCM before checking by division.
Record: Write the aim, materials, steps, observations and conclusion in your notebook. Follow teacher supervision for any classroom activity.
Quick practice with answers
Try each question before opening its solution.
Practice 1: Write 360 as a product of primes.
360 = 2³ × 3² × 5.
Practice 2: Find HCF(72,120) and LCM(72,120).
72 = 2³×3²; 120 = 2³×3×5. HCF = 24; LCM = 360.
Practice 3: Why does the proof of √3 irrational contradict lowest terms?
The proof shows that 3 divides both numerator and denominator, contradicting that the fraction was in lowest terms.
Revision checklist
- Can I explain the key idea without memorising a worked solution?
- Have I selected the right formula and written units where needed?
- Can I show each algebraic or geometric step clearly?
- Can I check my answer by substitution, estimation or a diagram?
Aligned to the Class X Mathematics curriculum for 2026–27 and the NCERT Mathematics textbook. Always confirm any school-specific internal assessment with your teacher.