Factors, Multiples and Divisibility

What you will learn

• How to tell whether a number is a factor or a multiple.
• How to use divisibility rules for 2, 3, 4, 5, 6, 9 and 10.
• How to list factor pairs systematically so no factors are missed.
• How to combine tests for divisibility by numbers such as 6 and 12.
• How to build a factor tree and write a number as a product of prime factors.
• How to show clear reasoning in GCSE-style answers.

Why this matters

Factors and multiples sit underneath lots of GCSE Maths topics. They help with simplifying fractions, finding common denominators, using ratios, factorising algebra and solving number problems.

They also help you work faster. A divisibility test can save you from doing long division when you only need to decide whether a number divides exactly.

Prior knowledge

You will be most comfortable with this lesson if you can already:

• multiply and divide whole numbers,
• use times tables up to 12 × 12,
• recognise odd and even numbers,
• understand that division can leave a remainder.

Clear explanation

A factor of a number divides into it exactly. For example, 6 is a factor of 42 because:

A multiple is what you get when you multiply a number by a whole number. The first few positive multiples of 6 are:

The words are connected. If 6 is a factor of 42, then 42 is a multiple of 6.

For full factor lists, work in pairs from 1 upwards. Stop when the next factor pair would repeat or cross over; that is how you know the list is complete.

Prime numbers and factor trees

A prime number has exactly two factors: 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11 and 13. A factor tree breaks a number down until only prime factors are left.

This tree shows:

Useful divisibility tests

• 2: the last digit is 0, 2, 4, 6 or 8.
• 3: the digit sum is divisible by 3.
• 4: the last two digits make a number divisible by 4.
• 5: the last digit is 0 or 5.
• 6: the number is divisible by both 2 and 3.
• 9: the digit sum is divisible by 9.
• 10: the last digit is 0.

Some divisibility tests combine two simpler tests. For example, a number is divisible by 6 if it is divisible by 2 and 3; it is divisible by 12 if it is divisible by 3 and 4.

Worked examples

Quick checks

Try these before you look at the practice answers. You will get instant feedback.

Practice questions

Answers and marking guidance

The practice answers are hidden under each question so you can try the work first. For this topic, marks usually come from systematic factor pairs, correct multiple lists, a valid divisibility test and factor-tree branches that keep splitting until only primes remain. When combining divisibility tests, say why the tests work together instead of just multiplying two rule numbers.

Common mistakes

• Mixing up factors and multiples: factors divide into a number; multiples are made by multiplying.
• Missing factor pairs: work systematically from 1 upwards so you do not skip a pair.
• Using only the last digit for divisibility by 3: for 3, use the sum of the digits.
• Stopping a factor tree too early: keep splitting until every branch ends in a prime number.
• Writing only a final answer for a show that question: GCSE marks often depend on the reasoning.

Extension challenge

Find a three-digit number that is divisible by 2, 3 and 5, but not divisible by 9. Explain how you know.

Exam-board guidance

Factors, multiples, primes and divisibility tests appear across GCSE Maths specifications as number-fluency skills. Expect exact division checks, factor or multiple lists, prime/composite language and short explanations that justify why a number does or does not divide exactly.

Next suggested lesson

Highest Common Factor and Lowest Common Multiple is the natural next step because it uses factors and multiples to solve comparison and grouping problems.

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