Multiples and Factors
Master multiples and factors for GCSE Maths with structured, exam-style practice. This Foundation resource covers finding multiples and factors and includes a complete mark scheme showing the steps examiners reward. Suitable for AQA, Edexcel and OCR. Free to download as a PDF. Factors divide into a number exactly; multiples come from its times table.
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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.
Challenge / Extension
Stretch yourself beyond the basics.
Topic overview
A factor of a number divides into it exactly, leaving no remainder. A multiple is the result of multiplying a number by a whole number, so multiples are what appear in that number's times table.
The two ideas are opposites of each other. Since \(4\) is a factor of \(12\), it follows that \(12\) is a multiple of \(4\). Keeping the direction straight is what most questions test.
Factors come in pairs. For \(24\), the pairs are \(1 \times 24\), \(2 \times 12\), \(3 \times 8\) and \(4 \times 6\), which gives all eight factors. Working through the pairs in order guarantees you find every one without missing any.
Revision notes
Listing factors in pairs
Start at \(1\) and work upwards, pairing each factor with its partner. Stop when the pairs begin to repeat.
For \(36\): \(1 \times 36\), \(2 \times 18\), \(3 \times 12\), \(4 \times 9\), \(6 \times 6\). That gives \(1, 2, 3, 4, 6, 9, 12, 18, 36\) — nine factors, since \(6\) pairs with itself.
Multiples
The multiples of a number are its times table, continuing forever. The multiples of \(7\) are \(7, 14, 21, 28, 35\) and so on.
Every number is a multiple of itself and of \(1\). A number has a finite list of factors but an infinite list of multiples.
Common factors and multiples
A common factor divides into two numbers, and a common multiple appears in both times tables.
The common factors of \(12\) and \(18\) are \(1, 2, 3\) and \(6\). The first common multiple of \(4\) and \(6\) is \(12\), which is their lowest common multiple.
Key points
- A factor divides exactly into a number.
- A multiple is the result of multiplying by a whole number.
- Factors come in pairs.
- A number has finitely many factors but infinitely many multiples.
- Every number is a factor and a multiple of itself.
- A square number has an odd number of factors.
Worked examples
Example 1
List all the factors of \(28\).
Working
Example 2
Write down the first five multiples of \(9\).
Working
Example 3
Find the common factors of \(20\) and \(30\).
Working
Common mistakes
Confusing factors with multiples.
Factors are smaller than or equal to the number; multiples are larger than or equal to it.
Missing factors by not working in pairs.
Listing pairs from 1 upwards guarantees a complete list.
Forgetting 1 and the number itself.
Both are always factors, and leaving them out loses marks.
Stopping the multiples list too early.
Multiples continue indefinitely, so a question asking for the first five wants exactly five.
Exam tips
- List factors in pairs from 1 upwards so none are missed.
- Check each factor divides exactly with no remainder.
- Write factors in ascending order in the final answer.
- For common factors, list both sets fully and then compare.
Key terms
- Factor
- A number that divides exactly into another.
- Multiple
- The result of multiplying a number by a whole number.
- Common factor
- A factor shared by two or more numbers.
- Factor pair
- Two factors that multiply to give the number.
Related topics
Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.