Number System and Real Numbers: Euclid's Lemma, HCF–LCM and Decimal Expansions
This unit tests Euclid's division lemma, the Fundamental Theorem of Arithmetic, HCF and LCM by prime factorisation, irrationality proofs and the terminating/recurring nature of decimal expansions. Expect short 1–2 mark items plus one 3-mark application.
Core rules
- Euclid's division lemma: for positive integers a and b, a = bq + r with 0 ≤ r < b. Repeating it on (b, r) gives the HCF.
- Fundamental Theorem of Arithmetic: every composite number has a unique prime factorisation apart from order.
- For two positive integers, HCF × LCM = a × b. This identity fails for three numbers.
- A rational p/q in lowest terms terminates exactly when q = 2^m·5^n; otherwise the decimal is non-terminating recurring.
- √p is irrational for every prime p. The proof is by contradiction, using: if p divides a² then p divides a.
Worked example
Find HCF and LCM of 96 and 404 by prime factorisation. 96 = 2^5 × 3; 404 = 2^2 × 101. HCF = 2^2 = 4, the lowest power of each prime common to both. LCM = 2^5 × 3 × 101 = 9696, the highest power of every prime present. Check: 4 × 9696 = 38784 and 96 × 404 = 38784, so the identity holds.
Traps that cost marks
- Applying HCF × LCM = product of numbers to three numbers.
- Testing q for the 2^m·5^n form before reducing the fraction: 3/6 terminates, but 6 is not of that form.
- Writing 0 ≤ r ≤ b instead of 0 ≤ r < b.
- Calling √9 irrational; only non-square integers have irrational square roots.