What are factors and how do you find them?

Factors are numbers that divide another number exactly without leaving a remainder. To find the factors of a number, you can start by dividing the number by the smallest prime number, which is 2, and continue with other prime numbers like 3, 5, 7, etc., until you reach the number itself. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.

What are multiples and how are they different from factors?

Multiples of a number are the products you get when you multiply that number by integers. Unlike factors, which are smaller or equal to the number, multiples are equal to or greater than the number. For instance, the first few multiples of 4 are 4, 8, 12, 16, and so on. Multiples continue infinitely, whereas factors are limited to the number itself and numbers smaller than it.

How can you determine the greatest common factor (GCF) of two numbers?

The greatest common factor (GCF) of two numbers is the largest number that divides both numbers exactly. To find the GCF, list all the factors of each number and identify the largest factor that appears in both lists. For example, for the numbers 18 and 24, the factors are 1, 2, 3, 6, 9, 18 and 1, 2, 3, 4, 6, 8, 12, 24, respectively. The GCF is 6.

What is the least common multiple (LCM) and how do you find it?

The least common multiple (LCM) of two numbers is the smallest multiple that is exactly divisible by both numbers. To find the LCM, list the multiples of each number and identify the smallest multiple common to both lists. For example, for 4 and 5, the multiples are 4, 8, 12, 16, 20, and 5, 10, 15, 20, respectively. The LCM is 20.

Why are factors and multiples important in the Cambridge Lower Secondary Mathematics curriculum?

Understanding factors and multiples is crucial in the Cambridge Lower Secondary Mathematics curriculum as they form the foundation for more advanced topics such as fractions, ratios, and algebra. Mastery of these concepts helps students develop problem-solving skills and prepares them for higher-level mathematics, where they will encounter these ideas in various contexts.

Why is "Saying that 1 is a prime number." a common mistake in Factors and Multiples?

Why it happens: 1 looks like it fits — it can only be divided by 1 and itself. How to avoid it: A prime has *exactly two* factors. 1 only has one factor (itself), so it is not prime.

Why is "Mixing up factors and multiples, e.g. saying '$30$ is a factor of $5$'." a common mistake in Factors and Multiples?

Why it happens: Both words are used a lot, and they sound similar. How to avoid it: Factors are *smaller* (or equal) — they divide into the number. Multiples are *larger* (or equal) — they come out of it.

Why is "Finding the LCM when the question asks for HCF, or vice versa." a common mistake in Factors and Multiples?

Why it happens: The acronyms are similar and the methods feel related. How to avoid it: HCF — common primes, **lowest** power. LCM — **every** prime, **highest** power. Check the size too: HCF smaller number, LCM larger number.

Why is "Stopping a factor tree too early, leaving a composite number at the end." a common mistake in Factors and Multiples?

Why it happens: A branch like '9' or '15' looks small enough to leave alone. How to avoid it: Keep splitting until every branch ends at a prime. 9 = 3 × 3 and 15 = 3 × 5.

Why is "Saying the HCF is 1 just because 1 appears in both factor lists, without checking for larger common factors." a common mistake in Factors and Multiples?

Why it happens: 1 is always a factor of every number, so it always shows up first. How to avoid it: Read the **whole** list and pick the **largest** value that appears in both — only call it 1 if there really is no other shared factor.

NumberCambridge Lower Secondary Mathematics (Checkpoint Exam Prep)

Factors and Multiples

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Factors and Multiples — Cambridge Lower Secondary Maths, Checkpoint

Revise factors, multiples, prime numbers and prime factorisation, then use them to find highest common factors and lowest common multiples — the building blocks behind fractions, ratio and so much more.

What you’ll learn

Mapped to the Cambridge Lower Secondary Mathematics curriculum framework.

Find all the factors and multiples of a whole number.
Identify prime numbers up to 100 and write any number as a product of its prime factors.
Find the highest common factor (HCF) of two or more numbers.
Find the lowest common multiple (LCM) of two or more numbers.

Factors and multiples

Factors divide into a number; multiples come out of it.

A factor of a whole number is any number that divides into it exactly, with no remainder. The factors of $12$ are $1, 2, 3, 4, 6$ and $12$ . Notice they come in pairs that multiply to give 12: $1 \times 12$ , $2 \times 6$ , $3 \times 4$ .

A multiple of a number is what you get when you multiply it by a whole number. The multiples of $4$ are $4, 8, 12, 16, 20, 24, \dots$ — they go on for ever.

Factors are a finite list; multiples carry on for ever.

The neat trick for finding all the factors of a number is to list pairs going up from $1$ . For $36$ : $1 \times 36$ , $2 \times 18$ , $3 \times 12$ , $4 \times 9$ , $6 \times 6$ . Stop when the two numbers in the pair meet (here at $6 \times 6$ ). The full list is $1, 2, 3, 4, 6, 9, 12, 18, 36$ .

Factors divide a number exactly; multiples are products of it.
Factors come in pairs that multiply to the original number.
Stop listing factor pairs when the two values meet.
Multiples of a number continue without end.

Prime numbers and prime factorisation

A prime has exactly two factors. Every whole number > 1 breaks into primes in just one way.

A prime number is a whole number greater than 1 with exactly two factors — 1 and itself. The first few primes are $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, \dots$ . Note that $2$ is the only even prime, and crucially 1 is not prime because it only has one factor.

A composite number is a whole number greater than 1 that is not prime — it has more than two factors. The Fundamental Theorem of Arithmetic says every composite number can be written as a product of primes in exactly one way (apart from the order). For example $60 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3 \times 5$ .

The easiest tool for finding the prime factorisation is a factor tree. Split the number into any factor pair, keep splitting each branch until only primes are left, then read them off the ends.

Keep splitting until every branch ends at a prime, then multiply them up.

The starting split does not matter — $60 = 4 \times 15$ or $60 = 2 \times 30$ both end at the same set of primes. Use indices to write repeated primes neatly, as in $60 = 2^2 \times 3 \times 5$ .

A prime has exactly two factors: 1 and itself.
$2$ is the only even prime; $1$ is not prime.
Every whole number > 1 factorises into primes uniquely.
A factor tree builds the prime factorisation step by step.

Highest Common Factor (HCF)

The HCF is the biggest factor two numbers have in common — most easily found from prime factor trees.

The Highest Common Factor (HCF) of two numbers is the largest number that divides into both of them. It is sometimes called the Greatest Common Divisor.

For small numbers you can list the factors of each and compare. The factors of $18$ are $1, 2, 3, 6, 9, 18$ . The factors of $24$ are $1, 2, 3, 4, 6, 8, 12, 24$ . The common factors are $1, 2, 3, 6$ — and the highest is $6$ . So $\text{HCF}(18, 24) = 6$ .

For larger numbers the reliable method is to prime factorise each number first, then multiply the primes they share — taking the lower power when a prime appears in both.

Take the primes the numbers share, at the lowest power that appears.

So $72 = 2^3 \times 3^2$ and $120 = 2^3 \times 3 \times 5$ . Both have $2^3$ (lowest power $2^3$ ) and $3$ (lowest power $3^1$ ). The HCF is $2^3 \times 3 = 24$ .

HCF = largest factor common to all the numbers.
List-and-compare works for small numbers.
For larger numbers, prime factorise first.
Take each shared prime to its lowest power.

Lowest Common Multiple (LCM)

The LCM is the smallest multiple two numbers share — take each prime to its highest power.

The Lowest Common Multiple (LCM) of two numbers is the smallest whole number that is a multiple of both.

You can find it by listing multiples until you spot the first one they share. The multiples of $4$ are $4, 8, 12, 16, 20, 24, \dots$ . The multiples of $6$ are $6, 12, 18, 24, \dots$ . The first they share is $12$ , so $\text{LCM}(4, 6) = 12$ .

For bigger numbers the prime-factorisation method is much faster. Prime factorise each number, then take every prime that appears, raised to its highest power.

Use every prime that appears, taken to its highest power in either number.

So $72 = 2^3 \times 3^2$ and $120 = 2^3 \times 3 \times 5$ . Take $2^3$ , $3^2$ and $5$ — the LCM is $2^3 \times 3^2 \times 5 = 360$ .

A useful link to remember: $\text{HCF} \times \text{LCM} =$ the product of the two original numbers. For 72 and 120: $24 \times 360 = 8640 = 72 \times 120$ .

LCM = smallest multiple that all the numbers share.
List-and-compare works for small numbers.
Take every prime, raised to its highest power.
HCF × LCM equals the product of the two numbers.

Using factors and multiples in problems

Word problems often translate straight into an HCF or LCM question.

Lots of real problems are HCF or LCM problems in disguise — the trick is spotting which one.

If two events happen on regular cycles and you want to know when they happen together, that is an LCM question. Two lighthouses flash every $12$ s and $15$ s — when do they next flash together? $\text{LCM}(12, 15) = 60$ s.

If you want to share things into equal groups without leftovers, or cut something into the biggest possible equal pieces, that is an HCF question. You have $24$ red and $36$ blue counters and want identical bags with both colours and nothing left over. The largest number of bags is $\text{HCF}(24, 36) = 12$ .

A quick test: "when do they meet" → LCM; "biggest equal share" → HCF.

Read the question slowly, decide which one fits, and use prime factorisation if the numbers are large.

When-things-happen-together questions → LCM.
Biggest-equal-share questions → HCF.
Prime factorise large numbers before combining.
Always check the answer makes sense in the original story.

Where you'll use this next

Factors and multiples power most of the topics that come after.

Strong factor and multiple work supports almost every later number topic:

Fractions use the LCM for adding and subtracting, and the HCF for simplifying.
Ratio problems often need an HCF to write a ratio in its simplest form.
Algebra factorises expressions by spotting common factors — exactly the same idea as HCF.
Indices and standard form use the same prime-power language.

Whenever a question mentions "common" — common factor, common multiple, common denominator — your factor work is doing the heavy lifting.

Fraction work uses LCM and HCF directly.
Ratio simplification is an HCF idea.
Algebra factorises by spotting common factors.
Indices and standard form rely on prime powers.

Sources: Cambridge Lower Secondary Mathematics curriculum framework. Last reviewed 2026-05-19.

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Step-by-step worked examples — Factors and Multiples

Step-by-step solutions for factors and multiples, written exactly the way a tutor would explain them at the board.

1Listing all the factors of a number
Getting started• factors, factor pairs
Question
List all the factors of $36$ .
Step-by-step solution
1. Step 1
  Work through factor pairs from 1 upwards: $1 \times 36$ , $2 \times 18$ , $3 \times 12$ , $4 \times 9$ , $6 \times 6$ .
2. Step 2
  Stop when the two numbers in the pair meet at $6 \times 6$ .
3. Step 3
  Write each value once, in order.
  $1,\ 2,\ 3,\ 4,\ 6,\ 9,\ 12,\ 18,\ 36$
Answer
$1, 2, 3, 4, 6, 9, 12, 18, 36$
2Common multiples by listing
Getting started• multiples, common multiples
Question
Write the first six multiples of $4$ and $6$ , then find their LCM.
Step-by-step solution
1. Step 1
  Multiples of 4 are $4, 8, 12, 16, 20, 24$ .
2. Step 2
  Multiples of 6 are $6, 12, 18, 24, 30, 36$ .
3. Step 3
  The first multiple that appears in both lists is the LCM.
  $\text{LCM}(4, 6) = 12$
Answer
$\text{LCM}(4, 6) = 12$
3Prime factorisation using a factor tree
Building confidence• prime factors, factor tree
Question
Write $84$ as a product of its prime factors.
Step-by-step solution
1. Step 1
  Split into any factor pair, e.g. $84 = 4 \times 21$ .
2. Step 2
  Break each branch down: $4 = 2 \times 2$ and $21 = 3 \times 7$ .
3. Step 3
  Collect the primes at the ends of the tree, in order.
  $84 = 2 \times 2 \times 3 \times 7$
4. Step 4
  Use index notation for repeated primes.
  $84 = 2^2 \times 3 \times 7$
Answer
$84 = 2^2 \times 3 \times 7$
4HCF from prime factorisation
Building confidence• hcf, prime factors
Question
Find the HCF of $48$ and $60$ .
Step-by-step solution
1. Step 1
  Prime factorise each number.
  $48 = 2^4 \times 3,\quad 60 = 2^2 \times 3 \times 5$
2. Step 2
  Pick out the primes that appear in both, each at the lowest power.
  $2: \text{ lowest power } 2^2.\quad 3: \text{ lowest power } 3^1.$
3. Step 3
  Multiply.
  $\text{HCF} = 2^2 \times 3 = 4 \times 3 = 12$
Answer
$\text{HCF}(48, 60) = 12$
5LCM in a real situation
Stretch• lcm, word problem
Question
Two lighthouses flash on regular cycles. The first flashes every $18$ seconds and the second every $24$ seconds. They flash together at midnight. When is the next time they flash together?
Step-by-step solution
1. Step 1
  'When do they happen together?' is an LCM question.
2. Step 2
  Prime factorise each cycle length.
  $18 = 2 \times 3^2,\quad 24 = 2^3 \times 3$
3. Step 3
  Take every prime that appears, raised to its highest power.
  $\text{LCM} = 2^3 \times 3^2 = 8 \times 9 = 72$
4. Step 4
  So they next flash together $72$ seconds after midnight — at $00{:}01{:}12$ .
Answer
$72$ seconds later — at $00{:}01{:}12$

Key Definitions and Keywords — Factors and Multiples

The important words for factors and multiples and what they mean — learn these so you can explain your thinking clearly.

Factor
Key word
A whole number that divides exactly into another whole number, leaving no remainder. The factors of $12$ are $1, 2, 3, 4, 6, 12$ .
Multiple
Key word
The result of multiplying a whole number by another whole number. The multiples of $5$ are $5, 10, 15, 20, \dots$
Factor pair
Two factors that multiply to give the original number, e.g. $3$ and $4$ form a factor pair of $12$ because $3 \times 4 = 12$ .
Prime number
Key word
A whole number greater than 1 with exactly two factors — itself and 1. The first primes are $2, 3, 5, 7, 11, 13, \dots$ . Note: 1 is not prime.
Composite number
A whole number greater than 1 that has more than two factors. Every composite number can be written as a product of primes.
Prime factorisation
Key word
Writing a number as a product of prime numbers. Every whole number greater than 1 has exactly one prime factorisation (apart from the order).
Example
$60 = 2^2 \times 3 \times 5$ .
Factor tree
A diagram used to find the prime factorisation of a number by splitting it into factor pairs until only primes remain.
Highest Common Factor (HCF)
Key word
The largest whole number that is a factor of two or more given numbers. Also called the Greatest Common Divisor.
Example
$\text{HCF}(18, 24) = 6$ .
Lowest Common Multiple (LCM)
Key word
The smallest whole number that is a multiple of two or more given numbers.
Example
$\text{LCM}(4, 6) = 12$ .
Coprime
Two numbers are coprime if their HCF is 1. They share no prime factors.
Example
$9$ and $20$ are coprime: $\text{HCF}(9, 20) = 1$ .
Index notation
Writing repeated multiplication with a small raised index, e.g. $2 \times 2 \times 2 = 2^3$ . Useful for short prime factorisations.
Divisible
A number is divisible by another if dividing leaves no remainder. $35$ is divisible by $5$ because $35 \div 5 = 7$ exactly.

Common Mistakes and Misconceptions — Factors and Multiples

The slip-ups students most often make with factors and multiples — and simple ways to avoid them.

✕Saying that 1 is a prime number.
Why it happens
1 looks like it fits — it can only be divided by 1 and itself.
How to avoid it
A prime has exactly two factors. 1 only has one factor (itself), so it is not prime.
✕Mixing up factors and multiples, e.g. saying ' $30$ is a factor of $5$ '.
Why it happens
Both words are used a lot, and they sound similar.
How to avoid it
Factors are smaller (or equal) — they divide into the number. Multiples are larger (or equal) — they come out of it.
✕Finding the LCM when the question asks for HCF, or vice versa.
Why it happens
The acronyms are similar and the methods feel related.
How to avoid it
HCF — common primes, lowest power. LCM — every prime, highest power. Check the size too: HCF $\le$ smaller number, LCM $\ge$ larger number.
✕Stopping a factor tree too early, leaving a composite number at the end.
Why it happens
A branch like '9' or '15' looks small enough to leave alone.
How to avoid it
Keep splitting until every branch ends at a prime. $9 = 3 \times 3$ and $15 = 3 \times 5$ .
✕Saying the HCF is 1 just because 1 appears in both factor lists, without checking for larger common factors.
Why it happens
1 is always a factor of every number, so it always shows up first.
How to avoid it
Read the whole list and pick the largest value that appears in both — only call it 1 if there really is no other shared factor.

Practice questions

Practice questions with step-by-step worked solutions. Try one before checking the method.

Practice quiz

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Factors and Multiples — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Factors and Multiples

Short Study Notes in the form of Slides

Detailed Notes

What you’ll learn

1Listing all the factors of a number

2Common multiples by listing

3Prime factorisation using a factor tree

4HCF from prime factorisation

5LCM in a real situation

Factor

Multiple

Factor pair

Prime number

Composite number

Prime factorisation

Factor tree

Highest Common Factor (HCF)

Lowest Common Multiple (LCM)

Coprime

Index notation

Divisible

✕Saying that 1 is a prime number.

✕Mixing up factors and multiples, e.g. saying '303030 is a factor of 555'.

✕Finding the LCM when the question asks for HCF, or vice versa.

✕Stopping a factor tree too early, leaving a composite number at the end.

✕Saying the HCF is 1 just because 1 appears in both factor lists, without checking for larger common factors.

Practice questions

Practice quiz

Assess your understanding

Factors and Multiples — frequently asked questions

✕Mixing up factors and multiples, e.g. saying ' $30$ is a factor of $5$ '.