Primes, factors, multiples, HCF and LCM

A foundational Edexcel topic. Both Foundation and Higher use prime factorisation; Higher applies HCF/LCM in algebraic contexts.

Vocabulary

• Factor of n: a positive integer that divides n exactly. Factors of 12: 1, 2, 3, 4, 6, 12.
• Multiple of n: any product n × k where k is a positive integer. Multiples of 4: 4, 8, 12, 16, ...
• Prime number: integer > 1 with exactly two factors (1 and itself). First primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
• Composite: integer > 1 that is not prime.

Note: 1 is neither prime nor composite.

Prime factorisation

Every integer > 1 can be written uniquely as a product of primes. Edexcel typically asks for index (power) form.

Use a factor tree (or repeated division by small primes):

60 = 2 × 30 = 2 × 2 × 15 = 2 × 2 × 3 × 5 = 2² × 3 × 5.

Standard form: 2² × 3 × 5.

Highest common factor (HCF)

HCF of m and n: largest integer dividing both.

Method (Edexcel recommended): list prime factorisations and take the lowest power of each shared prime.

HCF(60, 84): 60 = 2² × 3 × 5. 84 = 2² × 3 × 7. Common: 2² × 3 = 12. So HCF = 12.

Lowest common multiple (LCM)

LCM of m and n: smallest positive integer that is a multiple of both.

Method: take the highest power of every prime that appears in either.

LCM(60, 84): 2² × 3 × 5 × 7 = 4 × 3 × 5 × 7 = 420.

Useful identity: HCF × LCM = m × n. Check: 12 × 420 = 5040 = 60 × 84 ✓.

Edexcel exam tip

For a 3-mark "find the HCF and LCM of 36 and 90" question, lay out:

• Prime factorisation of each (M1 each)
• Compare and pick HCF and LCM with reasoning A1

A Venn diagram of prime factors (intersection = HCF, union = LCM) is a popular Edexcel mark-scheme presentation.

Common Edexcel question pattern

"Two buses leave a station at the same time. Bus A returns every 20 minutes and Bus B every 24 minutes. After how many minutes do they next leave together?"

Answer: LCM(20, 24). 20 = 2² × 5; 24 = 2³ × 3. LCM = 2³ × 3 × 5 = 120 minutes.