Factors, Multiples, Primes, and Structure in Number

The big idea

Every whole number greater than 1 has an internal structure. It can be built from prime factors in exactly one way. Understanding this structure reveals why some calculations simplify, when quantities can be grouped exactly, and how numbers relate to each other. Seeing inside a number is preparation for seeing inside an algebraic expression.

Why this matters for secondary maths

Factors, multiples, and primes are the architecture of number. Without this understanding, pupils are working with numbers as opaque objects. They can calculate with them but cannot see why certain simplifications work or why certain relationships hold.

For example, a pupil who simplifies $\frac{12}{18}$ to $\frac{2}{3}$ by ‘cancelling’ is performing a procedure. A pupil who understands that 12 = 2² × 3 and 18 = 2 × 3², and therefore sees the common factor of 6 = 2 × 3, is reasoning structurally. The second pupil can explain why the simplification works, apply the same reasoning to unfamiliar numbers, and, crucially, transfer the skill to algebraic fractions.

This section depends on:

the Laws of Arithmetic (Concept Section 2), which explain why factor structure behaves as it does.
the Area Model (Concept Section 7), which reveals factor pairs visually through rectangular arrays.
Division (Concept Section 8), which is the operation that defines factors and multiples.
Scaled Multiplication Tables (Concept Section 9), which show multiples as regular patterns.
Fractions, Decimals, and Percentages (Concept Section 12), which requires common factors for simplification.

The key idea connecting this section to the framework’s central thesis is that ‘seeing inside a number’, understanding how 6 and 9 share a common factor of 3, is the same reasoning as ‘seeing inside an expression’: recognising that $6 x + 9 = 3 (2 x + 3)$ . The process of factorising numbers is the conceptual precursor to factorising algebraic expressions.

Key representations

Rectangular arrays

A number’s factor pairs can be represented as rectangles. 12 can be arranged as 1 × 12, 2 × 6, or 3 × 4. Each arrangement is a rectangle with 12 unit squares. This connects directly to the area model and makes factor pairs visible as dimensions.

Strengths: Shows all factor pairs. Highlights that primes have only one rectangular arrangement (1 × p). Makes the connection between factors and area concrete.

Limitation: Becomes impractical for large numbers. Does not directly show prime factorisation.

Factor trees

A branching diagram where a number is repeatedly broken into factor pairs until all end points are prime. For example, 60 → 6 × 10 → (2 × 3) × (2 × 5) = 2² × 3 × 5.

Strengths: Shows the process of decomposition into primes. Makes the uniqueness of prime factorisation visible (different starting splits lead to the same prime factors). Parallels the decomposition of algebraic expressions into irreducible factors.

Limitation: Can become a mechanical procedure if not connected to the underlying idea. Pupils should understand what the tree reveals, not just how to draw it.

Venn diagrams for HCF and LCM

The prime factors of two numbers placed in a Venn diagram: common factors in the intersection, remaining factors in the outer regions. The HCF is the product of the intersection; the LCM is the product of all entries.

Strengths: Makes HCF and LCM visual and logical. Reveals why HCF × LCM = product of the two numbers.

Multiplication grids for multiples

Multiples shown as rows in a multiplication table (Concept Section 9) or as equally spaced points on a number line. Common multiples appear where rows align. This connects multiples to the scaled table structure and to the idea of periodic patterns.

Worked examples

Example 1: Finding all factor pairs

Find all the factors of 36.

Systematically: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6. We stop when the pairs start repeating (when the factor exceeds the square root of 36, which is 6). So the factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36.

Each factor pair corresponds to a rectangle with area 36, connecting back to the area model.

Example 2: Prime factorisation

Express 84 as a product of prime factors.

84 = 2 × 42 = 2 × 2 × 21 = 2 × 2 × 3 × 7 = 2² × 3 × 7.

No matter how we begin the factor tree (e.g. starting with 84 = 4 × 21 or 84 = 6 × 14), we always arrive at the same set of primes. This uniqueness is the Fundamental Theorem of Arithmetic: every whole number greater than 1 has exactly one prime factorisation.

Example 3: HCF and LCM using prime factorisation

Find the HCF and LCM of 24 and 36.

24 = 2³ × 3 36 = 2² × 3².

HCF: take the lowest power of each common prime factor. 2² × 3 = 12.

LCM: take the highest power of each prime factor present. 2³ × 3² = 72.

Check: 12 × 72 = 864 = 24 × 36. This relationship always holds.

Example 4: Simplifying a fraction using common factors

Simplify $\frac{42}{60}$ .

42 = 2 × 3 × 7 60 = 2² × 3 × 5.

The common factors are 2 and 3, so HCF = 6.

$\frac{42}{60} = \frac{42 \div 6}{60 \div 6} = \frac{7}{10}$ .

This is dividing numerator and denominator by the same number, which is dividing by $\frac{6}{6} = 1$ - the multiplicative identity (Concept Section 4). Simplifying fractions is not ‘cancelling’ in some vague sense; it is a precise application of the identity property.

The bridge to algebra

The reasoning used to find common factors of numbers is exactly the reasoning used to factorise algebraic expressions.

Factorising numbers → Factorising expressions:

Arithmetic: 6 + 9 = 3(2 + 3), because 3 is a common factor of 6 and 9.

Algebra: $6 x + 9 = 3 (2 x + 3)$ , because 3 is a common factor of $6 x$ and $9$ . The structure is identical.

HCF of numbers → HCF of algebraic terms:

Arithmetic: The HCF of 12 and 18 is 6.

Algebra: The HCF of $12 x^{2} y$ and $18 x y^{2}$ is $6 x y$ . The same process of identifying common factors extends to terms with variables.

Simplifying numerical fractions → Simplifying algebraic fractions:

Arithmetic: $\frac{12}{18} = \frac{2}{3}$ (dividing by the common factor 6).

Algebra: $\frac{12 x ^{2}}{18 x} = \frac{2 x}{3}$ (dividing by the common factor $6 x$ ). The reasoning is the same: identify what is common to numerator and denominator, and divide both by it.

Prime factorisation → Factorising quadratics:

Arithmetic: 35 = 5 × 7 (decomposing into prime factors).

Algebra: $x^{2} + 12 x + 35 = (x + 5) (x + 7)$ (decomposing into irreducible factors). The search for two numbers that multiply to give 35 and add to give 12 uses exactly the factor-pair reasoning practised with numbers.

The zero-product property:

If 5 × 7 = 35, we cannot have 5 × 7 = 0 (neither factor is zero). Conversely, if $ab = 0$ , then $a = 0$ or $b = 0$ (the zero-product property from Concept Section 4). This is the logical engine behind solving quadratics by factorising: if $(x + 5) (x + 7) = 0$ , then $x + 5 = 0$ or $x + 7 = 0$ .

The key message for teachers: Factorising in algebra is not a new skill. It is the same process of looking for common structure that pupils have been practising with numbers. The more confident pupils are at seeing inside numbers, the more naturally they will see inside expressions.

Key vocabulary

Term	Definition
Factor	A whole number that divides exactly into another whole number. 4 is a factor of 12 because 12 ÷ 4 = 3 with no remainder.
Multiple	The result of multiplying a number by any positive integer. The multiples of 4 are 4, 8, 12, 16, 20, …
Prime number	A whole number greater than 1 that has exactly two factors: 2, 3, 5, 7, 11, 13, …
Composite number	A whole number greater than 1 that has more than two factors. 12 is composite because it has factors 1, 2, 3, 4, 6, 12.
Prime factorisation	Expressing a number as a product of its prime factors: 60 = 2² × 3 × 5. Every whole number greater than 1 has a unique prime factorisation (the Fundamental Theorem of Arithmetic).
Highest common factor (HCF)	The largest number that is a factor of two or more numbers. HCF of 12 and 18 is 6.
Lowest common multiple (LCM)	The smallest number that is a multiple of two or more numbers. LCM of 4 and 6 is 12.
Divisibility	A number is divisible by another if division leaves no remainder. 24 is divisible by 6 because 24 ÷ 6 = 4.
Common factor	A factor shared by two or more numbers. 3 is a common factor of 12 and 18.
Factorise	To express a number or expression as a product of factors. 12 = 2² × 3 (number). $6 x + 9 = 3 (2 x + 3)$ (expression).

What we don’t say

Avoid	Why	Say instead
‘Cancel the top and bottom’	The word ‘cancel’ obscures the mathematics. It suggests that numbers disappear, rather than that we are dividing numerator and denominator by a common factor (which is dividing the fraction by 1 in the form $\frac{n}{n}$ ). ‘Cancelling’ also leads to errors like ‘cancelling’ terms that are added rather than multiplied.	‘Divide the numerator and denominator by their common factor.’ This names the operation precisely and connects to the multiplicative identity.
‘1 is a prime number’	1 is not prime. A prime number has exactly two distinct factors (1 and itself). The number 1 has only one factor (itself). Excluding 1 from the primes is essential for the uniqueness of prime factorisation.	‘A prime has exactly two factors: 1 and itself. Since 1 has only one factor, it is not prime.’
‘Just keep dividing by small numbers’ (for prime factorisation)	This describes a procedure without explaining what it achieves. Pupils should understand that they are decomposing a number into its fundamental building blocks.	‘Find a prime factor, divide, and repeat until the quotient is prime. You are breaking the number down into the prime building blocks that multiply together to make it.’
‘Factors are the small numbers’	Factors are not necessarily small. 50 is a factor of 100. The defining property is exact division, not size.	‘A factor of a number is any whole number that divides into it exactly.’

Common misconceptions and how to surface them

Misconception 1: Confusing factors and multiples.

Pupils frequently mix up ‘factors of 12’ and ‘multiples of 12’. Factors of 12 are the numbers that divide into 12 (1, 2, 3, 4, 6, 12, a finite list, all ≤ 12). Multiples of 12 are the results of multiplying 12 by whole numbers (12, 24, 36, … , an infinite list, all ≥ 12). Ask: ‘Is 4 a factor of 12 or a multiple of 12?’ Then: ‘Is 24 a factor of 12 or a multiple of 12?’ Pupils who hesitate on either need more work on the distinction.

Misconception 2: Believing 1 is prime.

This is common and persistent. Ask: ‘How many factors does 1 have? How many factors does 7 have?’ 1 has one factor (itself); 7 has two factors (1 and 7). Primes are defined by having exactly two distinct factors. This criterion excludes 1.

Misconception 3: Not finding all factor pairs systematically.

When asked to find the factors of 36, pupils often miss some. The systematic approach is to test 1, 2, 3, 4, 5, 6, … and stop when the factor meets or exceeds the square root. Ask: ‘Have you found all the factors of 24?’ and check whether they have 1, 2, 3, 4, 6, 8, 12, 24. Missing 8 is a common error.

Misconception 4: Thinking prime factorisation depends on the starting split.

Some pupils believe that starting a factor tree with 60 = 6 × 10 gives a different answer than starting with 60 = 2 × 30. Show both trees side by side and demonstrate that both end at 2² × 3 × 5. The uniqueness of prime factorisation is a fundamental property, not a coincidence.

Misconception 5: ‘Cancelling’ additive terms in fractions.

Pupils who have learned to ‘cancel’ sometimes write $\frac{3 + 6}{6} = \frac{3}{1} = 3$ , incorrectly ‘cancelling’ the 6. The correct simplification is $\frac{3 + 6}{6} = \frac{9}{6} = \frac{3}{2}$ . Only factors (multiplicative structures) can be divided out, not addends. This distinction becomes critical with algebraic fractions: ( $\frac{x + 6}{6} \neq = x$ . The language of ‘dividing by a common factor’ rather than ‘cancelling’ helps prevent this error.

Diagnostic questions

Question 1: A pupil says the HCF of 24 and 36 is 4. What have they done wrong, and what is the correct answer?

What this reveals: 4 is a common factor of 24 and 36, but it is not the highest. The correct HCF is 12. A pupil who identifies only 4 has found a common factor but has not been systematic. This distinguishes pupils who understand ‘highest’ from those who stop at the first common factor they notice. The prime factorisation method (24 = 2³ × 3, 36 = 2² × 3², HCF = 2² × 3 = 12) ensures completeness.

Question 2: Write $6 x + 15$ as a product of two factors. Explain how this is like simplifying $\frac{6}{15}$ .

What this reveals: $6 x + 15 = 3 (2 x + 5)$ , because 3 is the HCF of 6 and 15. Simplifying $\frac{6}{15}$ : the HCF of 6 and 15 is also 3, giving $\frac{6}{15} = \frac{2}{5}$ . A pupil who can explain both and see the connection, identifying common factors and extracting them, demonstrates that they understand factorising as a general structural skill, not as two separate procedures for numbers and expressions.

Progression spine

Stage	Key ideas	Notes
Primary (Y3–Y6)	Recognising multiples (times tables). Identifying factor pairs. Introduction to primes. Some divisibility tests.	Pupils typically arrive in Year 7 able to list some multiples and factors but without systematic methods or an understanding of prime factorisation.
Year 7	Systematic factor-finding. Prime factorisation using factor trees. HCF and LCM from prime factorisation. The Fundamental Theorem of Arithmetic. Connection between factors of numbers and simplifying fractions.	The emphasis is on understanding factor structure as revealing what numbers are ‘made of’, not just on procedural methods.
Years 8–11	Factorising algebraic expressions (common factor, difference of two squares, quadratics). Simplifying algebraic fractions. Algebraic proof involving divisibility. HCF and LCM in context (e.g. scheduling problems). The Fundamental Theorem as underpinning for work with surds and indices.	The transition from ‘finding factors of 12’ to ‘factorising $6 x + 9$ ’ should feel like the same skill in a new context. Pupils who have practised seeing inside numbers will see inside expressions.