Number Systems
This domain tests real numbers through Euclid's division lemma, the Fundamental Theorem of Arithmetic, HCF and LCM, and proofs that numbers such as √2, √3 and √5 are irrational.
Core rules
- Euclid's division lemma: for positive integers a and b, a = bq + r with 0 ≤ r < b. Repeated application is the division algorithm.
- Every composite number has a unique prime factorisation, ignoring the order of factors.
- For two positive integers, HCF(a, b) × LCM(a, b) = a × b. This identity does not extend directly to three numbers.
- A rational number p/q in lowest terms has a terminating decimal exactly when q = 2^m 5^n.
- Irrationality is proved by contradiction: assume √p = a/b in lowest terms, square both sides, and show that p divides both a and b, contradicting lowest terms.
Worked example
Find the HCF of 867 and 255. 867 = 255 × 3 + 102 255 = 102 × 2 + 51 102 = 51 × 2 + 0 The last non-zero remainder is 51, so HCF = 51. Check with the product rule: LCM = 867 × 255 ÷ 51 = 4335.
Traps
- Writing the lemma as a = bq − r or allowing r = b.
- Using HCF × LCM = product for three numbers.
- Cancelling in an irrationality proof without first assuming the fraction is in lowest terms.
- Treating a non-terminating but repeating decimal as irrational.