Number Systems and Real Numbers
This domain tests Euclid's division lemma, the fundamental theorem of arithmetic, HCF and LCM by prime factorisation, proofs that numbers such as √2 and √3 are irrational, and the decimal-expansion rule for rational numbers.
Core rules
- Euclid's lemma: for integers a and b > 0 there exist unique q and r with a = bq + r, 0 ≤ r < b. Repeating it gives the HCF.
- Fundamental theorem of arithmetic: every integer greater than 1 is a product of primes, unique up to order.
- For two positive integers, HCF(a, b) × LCM(a, b) = a × b. The rule fails for three numbers.
- A rational p/q in lowest terms has a terminating decimal exactly when q = 2^m 5^n; otherwise the decimal is non-terminating recurring.
- Irrationality pattern: suppose √p = a/b in lowest terms, square, deduce p divides both a and b, contradicting coprimality.
Worked example
Find the HCF and LCM of 96 and 404. Apply the lemma: 404 = 96×4 + 20; 96 = 20×4 + 16; 20 = 16×1 + 4; 16 = 4×4 + 0. The last non-zero remainder gives HCF = 4. Then LCM = (96 × 404)/4 = 9696.
Traps
Multiplying prime factors without taking the highest powers; applying the HCF–LCM product rule to three numbers; calling 0.101001000… recurring; using the terminating rule before reducing the fraction; forgetting to state that a and b are coprime in an irrationality proof.