Real Numbers and Number Systems
Real-number questions in this paper are short but proof-heavy: HCF and LCM by prime factorisation or Euclid's algorithm, classifying decimal expansions, and reproducing the standard irrationality proof. Marks are lost mostly through careless statements, not hard algebra.
Core rules
- Euclid's division lemma: for integers a and b (b > 0) there exist unique q, r with a = bq + r and 0 ≤ r < b. Repeat the step with (b, r) until r = 0; the last divisor is the HCF.
- Fundamental Theorem of Arithmetic: every integer greater than 1 factors uniquely into primes, apart from order. So HCF = product of the smallest power of each common prime, LCM = product of the greatest powers of all primes involved.
- For two positive integers only, HCF × LCM = a × b.
- A rational number p/q in lowest terms terminates if and only if q = 2^m · 5^n; otherwise the decimal repeats.
- Irrationality template: suppose √2 = p/q with HCF(p, q) = 1, square both sides, deduce 2 divides p, write p = 2k, deduce 2 divides q, contradicting coprimality.
Worked example
Find the HCF and LCM of 96 and 404, then verify the product rule. 96 = 2^5 × 3 and 404 = 2^2 × 101. Common prime power: 2^2, so HCF = 4. LCM = 2^5 × 3 × 101 = 9696. Check: 4 × 9696 = 38784 and 96 × 404 = 38784. The rule holds.
Traps
- Listing 1 as a prime in factorisation.
- Applying HCF × LCM = product to three numbers; it is false there.
- Testing the decimal rule before reducing the fraction: 6/15 looks non-terminating but equals 2/5.
- Quoting a calculator display as a proof of irrationality.
- Claiming 0.1010010001… (non-terminating, non-repeating) is rational.