Primes and prime factorisation
A prime number is a whole number greater than 1 with exactly two factors: 1 and itself. The primes begin 2, 3, 5, 7, 11, 13, … . Note that 1 is not prime, and 2 is the only even prime. Every whole number greater than 1 can be written as a product of primes in exactly one way.
Key idea
HCF (highest common factor) = the largest number that divides both. LCM (lowest common multiple) = the smallest number both divide into. For two numbers: HCF × LCM = product of the two numbers.
Finding HCF and LCM
Write each number as a product of prime factors. For the HCF, multiply the common prime factors (lowest powers). For the LCM, take every prime factor to its highest power.
Worked example
Find the HCF and LCM of 24 and 36.
24 = 23 × 3 and 36 = 22 × 32.
HCF = 22 × 3 = 12 (lowest powers of common primes).
LCM = 23 × 32 = 72 (highest powers).
Check: HCF × LCM = 12 × 72 = 864 = 24 × 36. ✓
Using them
HCF answers "how many equal groups?" questions (largest tile, most equal shares). LCM answers "when do events coincide?" questions (bells ringing together, buses meeting).
Remember
- 1 is not prime; 2 is the only even prime.
- HCF uses common factors; LCM uses all factors.
- HCF ≤ each number ≤ LCM.