1. What you must learn

• Euclid’s Division Lemma and Euclidean algorithm for HCF
• Fundamental Theorem of Arithmetic and prime factorisation
• HCF–LCM relationship
• Proofs of irrationality of √2, √3 and √5
• Terminating and non-terminating recurring decimal expansions

2. Euclid’s Division Lemma

For positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b. Repeatedly applying the lemma gives Euclid’s algorithm for finding HCF.

3. Fundamental Theorem of Arithmetic

Every composite number can be expressed as a product of primes, and this factorisation is unique apart from the order of the prime factors.

Example: 360 = 2³ × 3² × 5.

For two positive integers, HCF is obtained by taking the smallest powers of common primes, while LCM uses the greatest powers appearing in either factorisation.

4. HCF–LCM relationship

For positive integers a and b:

This relationship is useful as a check after factorisation.

5. Irrational numbers – proof pattern

To prove √2 is irrational, assume the opposite: √2 = p/q in lowest terms. Squaring gives p² = 2q², so p is even. Put p = 2k; then q is also even, contradicting that p/q was in lowest terms. Hence √2 is irrational.

The same contradiction method can be adapted for √3 and √5.

6. Decimal expansion of rational numbers

If p/q is in lowest form, its decimal expansion terminates exactly when the prime factorisation of q contains no primes other than 2 and 5.

7. Competency Case

A school measures two ropes as 84 m and 126 m and wants to cut both into equal maximum-length pieces without wastage.

1. Which mathematical concept is required?
2. Find the maximum possible length of each piece.
3. Find the total number of pieces.

Solution: HCF(84,126) = 42. Each piece is 42 m; therefore 84/42 + 126/42 = 5 pieces.

8. Exam Practice

1. Use Euclid’s algorithm to find HCF(1352, 168).
2. Show that 6ⁿ cannot end in 0 for any positive integer n.
3. Determine whether 77/160 has a terminating decimal expansion.
4. Prove that √3 is irrational.
5. If HCF of two numbers is 12 and their product is 8640, find their LCM.

9. Quick Revision

Division lemma → Euclid algorithm → prime factorisation → HCF/LCM → irrationality proofs → denominator test for terminating decimals.