Number System and Real Numbers
What the domain tests
This domain checks whether you can classify numbers, apply Euclid’s division lemma and algorithm, use the Fundamental Theorem of Arithmetic, and connect HCF and LCM with decimal expansions and irrationality proofs.
Core rules and concepts
- Euclid’s division lemma: for integers a and b > 0, there exist unique q and r with a = bq + r and 0 ≤ r < b.
- The Fundamental Theorem of Arithmetic says every integer greater than 1 has a unique prime factorisation, apart from order.
- For positive integers a and b, HCF(a, b) × LCM(a, b) = a × b.
- A rational number p/q has a terminating decimal exactly when q has no prime factors other than 2 and 5 after simplification.
- Irrationality is usually proved by contradiction: assume √2 = p/q in lowest terms, square, and derive that p and q are both even.
Worked example
Find the HCF and LCM of 84 and 120. Prime factorise: 84 = 2² × 3 × 7 and 120 = 2³ × 3 × 5. Take the lowest power of each common prime for HCF: 2² × 3 = 12. Take the highest power of every prime for LCM: 2³ × 3 × 5 × 7 = 840. Check: 12 × 840 = 10080 and 84 × 120 = 10080.
Traps that cost marks
Do not call every non-terminating decimal irrational: 1/3 = 0.333… is rational. Do not mix up HCF and LCM when the question asks for the smaller or larger value. In the √2 proof, state that p/q is in lowest terms and that a common factor of 2 contradicts that assumption. When using the HCF–LCM product rule, confirm both numbers are positive integers.