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Factors, Multiples & Primes

Factors, multiples and prime numbers describe how numbers are built from each other. They lead to two key exam skills: finding the Highest Common Factor (HCF) and the Lowest Common Multiple (LCM).

45 min Video by Maths Genie Pearson Edexcel GCSE Maths (1MA1)

The big picture

Prime factorisation is the DNA of a whole number. Once a number is split into its unique prime factors, HCF and LCM become structured comparisons instead of guesswork. Edexcel tests both the calculation and the interpretation of which measure a context requires.

Factors, Multiples and PrimesWatch the walkthrough, then read the notes below.

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What you'll learn

Find the factors and multiples of a number
Identify prime numbers
Write a number as a product of prime factors
Find the HCF and LCM of two numbers

The key words

A $factor$ divides exactly into a number (factors of 12: 1, 2, 3, 4, 6, 12). A $multiple$ is in that number’s times table (multiples of 4: 4, 8, 12, …). A $prime$ number has exactly two factors: 1 and itself (2, 3, 5, 7, 11, …).

Why this works

A factor divides exactly; a multiple belongs to a times table. Testing divisibility is usually faster than listing every possibility.

Tip — 1 is NOT prime (it has only one factor). 2 is the only even prime.

Prime factors, HCF and LCM

Every number can be written as a product of primes using a factor tree, e.g. $12 = 2^{2} \times 3$ . From the prime factors you can find the HCF (multiply the primes they share) and the LCM (multiply each prime the most times it appears in either number).

12 = 2^{2} \times 3

18 = 2 \times 3^{2}

2HCF = shared primes =

2 \times 3 = 6

3LCM =

2^{2} \times 3^{2} = 36

AnswerHCF

= 6

, LCM

= 36

Why this works

The fundamental theorem of arithmetic guarantees one prime factorisation for every integer greater than 1, apart from the order of the factors.

Choosing HCF or LCM in a problem

Use HCF when splitting or grouping quantities into the largest identical units. Use LCM when repeated events must meet again or when finding the smallest common quantity.

18 = 2 im es 3^{2}

and

24 = 2^{3} im es 3

2For the LCM, take the highest power of each prime:

2^{3} im es 3^{2}

8 im es 9 = 72

Answer

72

seconds

Why this works

The words “largest equal groups” point to HCF; “first time together again” points to LCM.

Think like an examiner

A prime factor tree must finish with prime numbers at every branch end.
When the question says “write as a product of prime factors,” use index notation where factors repeat.
In context questions, state what the HCF or LCM represents and include the correct unit.

Remember these

12 = 2^{2} \times 3

Product of prime factors (from a factor tree).

HCF \times LCM = a \times b

A handy check for two numbers a and b.

Watch out for these

Saying 1 is a prime number.

A prime has exactly two factors; 1 has only one, so it is not prime.

Mixing up HCF and LCM.

HCF is the biggest number that divides both (smaller); LCM is the first number both go into (larger).

Stretch yourself

Two numbers have HCF $12$ and LCM $420$ . One number is $60$ . Find the other number.

Hint — For two positive integers, product of the numbers = HCF $im es$ LCM.

Questions students ask

Is 1 a prime number?

No. A prime number has exactly two positive factors, 1 and itself. The number 1 has only one positive factor.

Can I use a calculator for HCF and LCM?

A calculator may help with division, but Edexcel questions often award marks for prime-factor method, so show the factorisation and selection of powers.

Key takeaways

Factors divide in; multiples are in the times table.
Primes have exactly two factors; 1 is not prime.
Use prime factors to find HCF (shared) and LCM (all, most times).

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