Launching your learning experience…
Every whole number can be broken down or built up from smaller ones. This guide shows you factors and multiples, prime numbers, how to split a number using a factor tree, and how to find the HCF and LCM of a pair of numbers.
Mapped to the Cambridge Lower Secondary Mathematics curriculum framework.
Factors divide a number exactly; multiples are what you reach by counting in a number.
Two words sit at the heart of this topic, and they pull in opposite directions.
A factor of a number divides into it exactly, with nothing left over. The factors of 12 are — each one divides 12 with no remainder.
A multiple of a number is what you get by multiplying it by a whole number. The multiples of 12 are — the answers in its times table.
A common factor or multiple is one that two numbers share.
Once you can list factors and multiples for one number, you can compare two numbers and look for the ones they have in common.
A common factor is a number that is a factor of both numbers. The factors of 8 are and the factors of 12 are . The ones in both lists — — are the common factors.
A prime number has exactly two factors: 1 and itself.
A prime number is a whole number greater than 1 that has exactly two factors — 1 and the number itself. Nothing else divides into it.
So 7 is prime because its only factors are 1 and 7. But 9 is not prime, because 3 divides into it as well, giving it three factors: . A number with more than two factors is called a composite number.
Every number can be written as a product of prime numbers.
Here is a powerful idea: every whole number bigger than 1 can be built by multiplying prime numbers together. Writing a number this way is called prime factorisation.
The friendliest method is a factor tree. Split the number into any two factors, then keep splitting until every branch ends on a prime.
The green numbers at the bottom — — are the prime factors. So , which is tidier still as .
The HCF is the biggest shared factor; the LCM is the smallest shared multiple.
Two special values describe how a pair of numbers relate.
The highest common factor (HCF) is the biggest number that divides into both. For 8 and 12, the common factors are , so the HCF is 4.
The lowest common multiple (LCM) is the smallest number that both divide into. For 4 and 6, the common multiples are , so the LCM is 12.
Factors, primes, HCF and LCM power up fractions, algebra and problem solving.
These ideas turn up again and again as you move forward:
If a fractions topic later feels shaky, it is often the HCF or LCM underneath that needs a quick refresh. These number-building skills quietly support a huge amount of maths to come.
Sources: Cambridge Lower Secondary Mathematics curriculum framework (Stage 7). Last reviewed 2026-05-19.
Step-by-step solutions for factors, multiples and prime, written exactly the way a tutor would explain them at the board.
Question
List all the factors of 18.
Step-by-step solution
Step 1
Search in pairs, starting from 1: .
Step 2
Next, and .
Step 3
4 and 5 do not divide 18 exactly, so the pairs are complete.
Step 4
Write every number from the pairs in order.
Answer
The factors of 18 are .
Question
Write the first five multiples of 7.
Step-by-step solution
Step 1
Multiply 7 by the whole numbers 1, 2, 3, 4 and 5 in turn.
Step 2
Work through the 7 times table.
Question
Is 51 a prime number? Explain how you decide.
Step-by-step solution
Step 1
A prime has exactly two factors. Test small numbers to see if anything else divides 51.
Step 2
Check 3: , with no remainder.
Question
Write 60 as a product of its prime factors.
Step-by-step solution
Step 1
Split 60 into any factor pair, such as .
Step 2
Split each branch: and .
Question
Find the highest common factor and the lowest common multiple of 24 and 36.
Step-by-step solution
Step 1
Write each number as a product of primes.
The important words for factors, multiples and prime and what they mean — learn these so you can explain your thinking clearly.
A whole number that divides exactly into another number, leaving no remainder.
Example
3 is a factor of 12 because .
The result of multiplying a number by a whole number — the answers in its times table.
Example
20 is a multiple of 5 because .
Two factors that multiply together to give a number, used to list factors without missing any.
Example
For 12, the factor pairs are , and .
A number that is a factor of two or more given numbers.
Example
4 is a common factor of 8 and 12.
A number that is a multiple of two or more given numbers.
Example
24 is a common multiple of 4 and 6.
A whole number greater than 1 that has exactly two factors — 1 and itself.
Example
13 is prime because only 1 and 13 divide it.
A whole number greater than 1 that has more than two factors.
Example
9 is composite because and all divide it.
A factor of a number that is itself a prime number.
Example
The prime factors of 12 are 2 and 3.
Writing a number as a product of its prime factors, such as .
A diagram that splits a number into factors over and over until every branch ends on a prime.
The largest number that is a factor of two or more given numbers.
Example
The HCF of 8 and 12 is 4.
The smallest number that is a multiple of two or more given numbers.
Example
The LCM of 4 and 6 is 12.
The slip-ups students most often make with factors, multiples and prime — and simple ways to avoid them.
Why it happens
The two words sound similar and are easy to confuse under pressure.
How to avoid it
Factors divide into a number and stay small; multiples are what you reach by counting up in it.
Why it happens
1 only divides by itself, so it feels like it has 'no other factors'.
How to avoid it
A prime needs exactly two different factors. 1 has only one factor, so it is not prime.
Why it happens
Factors are searched randomly instead of in an ordered way.
How to avoid it
Work in factor pairs from 1 upward: , , — stop when the pair meets.
Why it happens
'Highest' and 'lowest' get mixed up, especially in word problems.
How to avoid it
HCF is the biggest factor shared; LCM is the smallest multiple shared. Match 'factor' with HCF and 'multiple' with LCM.
Why it happens
It looks finished once the numbers are small.
How to avoid it
A branch is only finished when it reaches a prime. Keep splitting — 4 becomes .
All rights reserved
©2026 Tutopiya
All resources on this platform are independently created by Tutopiya and have no endorsement from the International Baccalaureate Organization.
A neat way to keep them apart: factors are smaller than or equal to the number and there are only a few of them, while multiples are equal to or bigger than the number and the list goes on forever.
To list factors without missing any, search in pairs. For 12: , , — work upwards until the pair meets in the middle.
A common multiple works the same way the other direction. The multiples of 4 are and the multiples of 6 are The shared ones — — are the common multiples.
Spotting common factors and multiples is the gateway to the HCF and LCM later in this guide, and it is also how you simplify fractions and find common denominators.
Two facts trip people up. First, 1 is not prime — it has only one factor, not two. Second, 2 is prime, and it is the only even prime, because every other even number can also be divided by 2.
The primes up to 20 are . They are worth knowing well.
A wonderful fact: it does not matter how you start the splitting. Begin 36 as or as and you always end on exactly the same set of primes.
For small numbers, listing works fine. For larger numbers, prime factorisation is faster. Write both numbers as products of primes, then for the HCF multiply the primes they share, and for the LCM take every prime, at its highest power.
The HCF helps you simplify fractions; the LCM helps you add fractions and solve "when do two events meet again" problems.
Answer
The first five multiples of 7 are .
Step 3
51 has the factors 1, 3, 17 and 51 — that is four factors, not two.
Answer
No — 51 is not prime, because .
Step 3
Every branch now ends on a prime. Collect them all.
Step 4
Write the repeated prime using index notation.
Answer
Step 2
For the HCF, multiply the primes they share, at the lowest power: two 2s and one 3.
Step 3
For the LCM, take every prime at its highest power: three 2s and two 3s.
Answer
HCF and LCM .