What Are Prime Factors?

Prime factors are the prime numbers that multiply together to make a number.

Formula: 12 = 2 × 2 × 3 = 2² × 3

Example: 30 = 2 × 3 × 5 60 = 2² × 3 × 5

Factor Trees

Split the number into factors, then split those until only primes (circles) remain.

HCF (Highest Common Factor)

The largest number that divides into both numbers.

Example: HCF of 12 & 18 is 6.

LCM (Lowest Common Multiple)

The smallest number in both times tables.

Example: LCM of 4 & 6 is 12.

Prime Numbers

A number with exactly two factors: 1 and itself.

Example: 2, 3, 5, 7, 11, 13, 17, 19...

Checking with Multiplication

To check your prime factorisation, multiply all the primes together. It should equal the original number.

Example: 2² × 3 × 5 = 4 × 3 × 5 = 60 ✓

Venn Diagram Method

Put prime factors in a Venn diagram. HCF is the intersection (center). LCM is the union (everything inside).

Example: 12: 2, 2, 3 18: 2, 3, 3 Middle: 2, 3 (HCF=6) Outer: 2 & 3 (LCM=2×2×3×3=36)

Product of Primes

Write any number as a product of its prime factors using index notation.

Formula: 360 = 2³ × 3² × 5

Example: 360 ÷ 2 = 180... keeps going until 1.

HCF × LCM Rule

For any two numbers a and b: HCF × LCM = a × b.

Formula: HCF × LCM = Product of Numbers

Example: a=6, b=10 HCF=2, LCM=30 2 × 30 = 60; 6 × 10 = 60 ✓

Exam Tip

If asked for HCF of large numbers, always use the Venn Diagram method to avoid missing factors.

HCF of Three Numbers

Find the prime factors of all three. The HCF is the product of primes shared by ALL three lists.

Example: 12, 18, 30: All share a 2 and a 3. HCF = 6.

Problem Solving

HCF is used for "cutting into equal pieces". LCM is used for "when things happen at the same time again".