Number Systems
Number Systems is where the HSLC paper tests your grip on divisibility, prime factorisation, HCF and LCM, and the exact classification of decimals. Questions are usually short but hinge on one precise definition.
Core rules
- Euclid's division lemma: for positive integers a and b, a = bq + r with 0 ≤ r < b. Repeatedly replacing (a, b) by (b, r) gives HCF(a, b).
- Fundamental Theorem of Arithmetic: every integer greater than 1 is a unique product of primes, apart from order.
- For two positive integers, HCF × LCM = product of the numbers. This identity does not extend to three numbers.
- A rational number p/q in lowest terms has a terminating decimal exactly when q = 2^m5^n; otherwise the decimal repeats.
Worked example
Find HCF(867, 255). 867 = 255 × 3 + 102; 255 = 102 × 2 + 51; 102 = 51 × 2 + 0. The last non-zero remainder is 51, so HCF = 51. Check: 867 = 17 × 51 and 255 = 5 × 51, with 17 and 5 coprime.
Traps
Do not write a remainder equal to the divisor: the lemma demands 0 ≤ r < b. In “prove √2 is irrational”, the contradiction must come from a and b being coprime. When a question asks for the smallest number that leaves remainder 3 on division by 8 and 12, you need LCM(8,12)+3 = 27, not HCF.