Number Systems and Real Numbers
This section covers Number Systems and Real Numbers. The exam tests Euclid’s division algorithm, the Fundamental Theorem of Arithmetic, HCF and LCM by prime factorisation, irrationality proofs, and decimal expansions of rational numbers.
Core rules
- Euclid’s lemma: for positive integers a and b, write a = bq + r with 0 ≤ r < b; then HCF(a,b) = HCF(b,r). Repeat until r = 0.
- Every composite number has a unique prime factorisation apart from the order of factors.
- HCF is the product of the smallest power of each common prime factor; LCM is the product of the greatest power of each prime present.
- For two positive integers, HCF × LCM = product of the numbers.
- A rational number p/q in lowest terms has a terminating decimal exactly when q = 2^m5^n; otherwise the decimal is non-terminating recurring.
- Irrationality is proved by contradiction: assume √p = a/b in lowest terms, square, and force a common factor.
Worked example
Find the HCF and LCM of 84 and 120. Prime factorise: 84 = 2²×3×7 and 120 = 2³×3×5. HCF = 2²×3 = 12. LCM = 2³×3×5×7 = 840. Check: 12×840 = 10080 and 84×120 = 10080.
Traps
Do not use HCF × LCM = product for more than two numbers. In irrationality proofs, state that a/b is in lowest terms before deriving the contradiction. For terminating decimals, reduce the fraction first: 6/15 terminates because it reduces to 2/5, although the original denominator contains 3.