Primes, Factors and Multiples

A firm understanding of the relationships between numbers is essential throughout GCSE Mathematics — from simplifying fractions to solving problems involving timing or measurement. This topic covers the key vocabulary and methods for analysing the structure of whole numbers.

Term	Definition	Example
Factor	A whole number that divides exactly into another with no remainder.	Factors of 12: 1, 2, 3, 4, 6, 12
Multiple	A number obtained by multiplying a given number by a positive integer.	Multiples of 4: 4, 8, 12, 16, 20 ...
Prime number	A number with exactly two factors: 1 and itself. Note: 1 is NOT prime.	2, 3, 5, 7, 11, 13, 17, 19 ...
Composite number	A positive integer with more than two factors (i.e. not prime and not 1).	4, 6, 8, 9, 10, 12 ...

Prime Factorisation

Prime factorisation expresses any composite number as a product of its prime factors. The Unique Factorisation Theorem (also called the Fundamental Theorem of Arithmetic) guarantees that every integer greater than 1 has exactly one prime factorisation — the order of the factors may vary but the set of prime factors is unique.

The most common method is the factor tree: split the number into any two factors, then continue splitting each composite factor until only primes remain. Below is the factor tree for 60.

\[ 60 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3 \times 5 \]

Write the prime factors in index form (ascending order of prime base) — this is the expected form in GCSE answers.

HCF and LCM from Prime Factorisations

Once two numbers are expressed as products of prime factors, finding the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) becomes systematic:

HCF: take each common prime factor raised to its lowest power.
LCM: take every prime factor (from either number) raised to its highest power.

Worked Examples

Express 180 as a product of prime factors in index form.

Divide repeatedly by the smallest prime that divides exactly, recording each step:

\[ 180 \div 2 = 90 \] \[ 90 \div 2 = 45 \] \[ 45 \div 3 = 15 \] \[ 15 \div 3 = 5 \]

5 is prime — stop here. Collecting all the divisors:

\[ 180 = 2 \times 2 \times 3 \times 3 \times 5 = 2^2 \times 3^2 \times 5 \]

Find the HCF of 48 and 60.

First, write each number as a product of prime factors:

\[ 48 = 2^4 \times 3 \] \[ 60 = 2^2 \times 3 \times 5 \]

Identify the common prime factors: both contain \(2\) and \(3\).

Take the lowest power of each common factor: \( 2^2 \) and \( 3^1 \).

\[ \text{HCF}(48, 60) = 2^2 \times 3 = 4 \times 3 = 12 \]

Find the LCM of 12 and 18.

\[ 12 = 2^2 \times 3 \] \[ 18 = 2 \times 3^2 \]

List all prime factors that appear in either number: \(2\) and \(3\).

Take the highest power of each: \( 2^2 \) and \( 3^2 \).

\[ \text{LCM}(12, 18) = 2^2 \times 3^2 = 4 \times 9 = 36 \]

Check: 36 is divisible by both 12 (\( 36 \div 12 = 3 \)) and 18 (\( 36 \div 18 = 2 \)). \( \checkmark \)

Key Takeaways

Every integer greater than 1 has a unique prime factorisation — this is guaranteed by the Unique Factorisation Theorem.
A factor tree splits a composite number into factor pairs repeatedly until all leaves are prime; the product of the leaves is the prime factorisation.
Write prime factorisations in index form in ascending order of prime base, e.g. \( 2^3 \times 3^2 \times 5 \).
To find the HCF: take common prime factors at their lowest powers.
To find the LCM: take all prime factors at their highest powers.
HCF is useful when splitting into equal groups or finding the largest tile/tile size; LCM is useful for finding the first time two events coincide.