Factors, multiples, primes, HCF and LCM
📖Definition— Definitions
- Factor of n — divides n exactly. Factors of 12: 1, 2, 3, 4, 6, 12.
- Multiple of n — any product n × k. Multiples of 4: 4, 8, 12, 16…
- Prime — exactly two distinct positive factors (1 and itself). 1 is not prime. 2 is the only even prime.
- Composite — has more than two factors.
Prime factorisation
Every integer ≥ 2 can be written as a unique product of primes (Fundamental Theorem of Arithmetic).
Use a factor tree: 60 → 2 × 30 → 2 × 2 × 15 → 2 × 2 × 3 × 5 = 2² × 3 × 5.
Highest Common Factor (HCF)
The largest number that divides all the given numbers.
Method 1: list factors of each, pick the largest common one. Method 2 (better): prime factorisation, then take the lowest power of each common prime.
Example: HCF of 60 and 84. 60 = 2² × 3 × 5 84 = 2² × 3 × 7 HCF = 2² × 3 = 12.
Lowest Common Multiple (LCM)
The smallest positive number that all the given numbers divide into.
From prime factorisation, take the highest power of every prime that appears.
Example: LCM of 60 and 84. 60 = 2² × 3 × 5 84 = 2² × 3 × 7 LCM = 2² × 3 × 5 × 7 = 420.
Useful identity
For two numbers a and b: a × b = HCF(a, b) × LCM(a, b). Check: 60 × 84 = 5040 = 12 × 420. ✓
Common CCEA contexts
- Word problems: "Two buses leave at 9 am — one every 12 min, the other every 18 min. When do they next leave together?" → LCM(12, 18) = 36 → 9:36 am.
- Cake/sweets sharing: "What is the largest number of bags that can be filled identically from 24 sweets and 36 chocolates?" → HCF(24, 36) = 12.
Common CCEA exam tip
Always show your prime factorisation — that earns the M1 even if you make a slip computing the HCF/LCM.
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