Factors, Multiples and Primes (HCF & LCM)

Factors, multiples and primes

• A factor of a number divides into it exactly (no remainder). Factors of 12: 1, 2, 3, 4, 6, 12.
• A multiple is the result of multiplying a number by a whole number. Multiples of 5: 5, 10, 15, 20, …
• A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11, 13…). Note: 1 is NOT prime (it has only one factor). 2 is the only even prime.

Prime factorisation

Every whole number can be written as a product of prime factors. Use a factor tree: keep splitting into factors until every branch ends in a prime.

Example — 60:

60 = 6 × 10
6 = 2 × 3,  10 = 2 × 5

So 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5.

Highest Common Factor (HCF)

The HCF is the largest number that divides into two (or more) numbers.

• Method: write each number as a product of primes; multiply the common prime factors.

Example: HCF of 24 (= 2³ × 3) and 60 (= 2² × 3 × 5).

• Common: 2² and 3 → HCF = 4 × 3 = 12.

Lowest Common Multiple (LCM)

The LCM is the smallest number that is a multiple of both.

• Method: multiply the highest power of every prime that appears in either number.

Example: LCM of 24 (= 2³ × 3) and 60 (= 2² × 3 × 5).

• Take 2³, 3 and 5 → LCM = 8 × 3 × 5 = 120.

Venn diagram method

Put the shared prime factors in the overlap, the rest in each circle.

• HCF = product of the overlap. LCM = product of everything in the diagram.

Worked example

Find the HCF and LCM of 18 and 24.

• 18 = 2 × 3²; 24 = 2³ × 3.
• HCF = common = 2 × 3 = 6.
• LCM = 2³ × 3² = 8 × 9 = 72. ✓

Common mistakes

• Thinking 1 is prime — it isn't.
• Swapping HCF and LCM — HCF is the largest factor (smaller number); LCM is the smallest multiple (larger number).
• Missing prime factors in the factor tree.

Exam tips

• Write numbers as products of primes first — it makes HCF/LCM straightforward.
• HCF = common primes; LCM = highest power of each prime.
• Remember 2 is the only even prime, and 1 is not prime.

Key facts to remember

• Factor divides exactly; multiple is in the times table; prime has exactly two factors (1 not prime).
• Write numbers as products of primes (factor tree).
• HCF = product of common primes (largest factor); LCM = product of the highest power of each prime (smallest multiple).