AQA · GCSE · Mathematics
Course version: 8300 · Version from Sept 2015
Structure and calculation: use the concepts and vocabulary of prime numbers, factors (divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple, prime factorisation, including using product notation and the unique factorisation theorem
This topic builds the vocabulary of number structure.
- 4 key terms
Study notes
From the checked teaching notes for this specification point.
The idea
This topic builds the vocabulary of number structure. A prime number has exactly two factors, 1 and itself, so 2, 3, 5, 7, 11 and 13 are prime, while 1 is not prime. A factor (divisor) of a number divides it exactly; a multiple is the result of multiplying it by an integer. Common factors are shared by two or more numbers, and the highest common factor (HCF) is the largest of these. Common multiples are shared multiples, and the lowest common multiple (LCM) is the smallest positive one. Prime factorisation writes a number as a product of primes, for example 360 = 2³ × 3² × 5. The unique factorisation theorem states that every integer greater than 1 has a prime factorisation that is unique apart from the order of the factors.
What earns the marks
- Defines a prime number as having exactly two distinct factors, 1 and itself, and identifies primes correctly.
- Lists factors (divisors) and multiples of a given number and distinguishes between the two ideas.
- Finds common factors and common multiples of two or more numbers and selects the highest common factor or lowest common multiple as required.
- Writes a composite number as a product of primes using index notation, for example 360 = 2³ × 3² × 5.
- Uses prime factorisation to find the HCF and LCM of two numbers by comparing prime factors.
- States that the prime factorisation of an integer greater than 1 is unique apart from the order of the factors.
Common mistakes and how to put them right
Treating 1 as a prime number.
Put it right: 1 has only one factor, so it is not prime; the smallest prime is 2.
Confusing factors with multiples.
Put it right: Factors divide a number exactly, while multiples are obtained by multiplying the number by integers.
Stopping a factor tree before all branches are prime.
Put it right: Continue until every branch ends in a prime, then write the product with indices.
Choosing the LCM as the product of the two numbers without checking for shared factors.
Put it right: For 6 and 8 the LCM is 24, not 48, because 6 and 8 share the factor 2.
Exam tips
- Write a factor tree or repeated division clearly so the prime factors can be checked, then collect repeated primes into index form.
- For HCF and LCM questions, write both prime factorisations in index form and compare powers of each prime.
- Check whether a question asks for the highest common factor or the lowest common multiple before giving a final answer.
Try the lesson
The start of the Structure and calculation: use the concepts and vocabulary of prime numbers, factors (divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple, prime factorisation, including using product notation and the unique factorisation theorem lesson, made from checked teaching notes for this specification point.
Have a go first
Which of these numbers is a prime number?
Listen
This topic builds the vocabulary of number structure. A prime number has exactly two factors, 1 and itself, so 2, 3, 5, 7, 11 and 13 are prime, while 1 is not prime. A factor (divisor) of a number divides it exactly; a multiple is the result of multiplying it by an integer. Common factors are shared by two or more numbers, and the highest common factor (HCF) is the largest of these. Common multiples are shared multiples, and the lowest common multiple (LCM) is the smallest positive one. Prime factorisation writes a number as a product of primes, for example 360 = 2³ × 3² × 5. The unique factorisation theorem states that every integer greater than 1 has a prime factorisation that is unique apart from the order of the factors. A strong answer earns marks for these points. Defines a prime number as having exactly two distinct factors, 1 and itself, and identifies primes correctly. Lists factors (divisors) and multiples of a given number and distinguishes between the two ideas. Finds common factors and common multiples of two or more numbers and selects the highest common factor or lowest common multiple as required. Writes a composite number as a product of primes using index notation, for example 360 = 2³ × 3² × 5. A common slip: Treating 1 as a prime number. Instead, 1 has only one factor, so it is not prime; the smallest prime is 2. Exam tip: Write a factor tree or repeated division clearly so the prime factors can be checked, then collect repeated primes into index form.
Key words
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Prime numbers
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Factors (divisors)
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Multiples
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Highest common factor
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See every definition
- Prime numbers
- A number that has exactly two factors, 1 and itself, so 2, 3, 5, 7, 11 and 13 are prime, while 1 is not prime.
- Factors (divisors)
- A factor of a number divides it exactly.
- Multiples
- A multiple is the result of multiplying a number by an integer.
- Highest common factor
- Common factors are shared by two or more numbers, and the highest common factor is the largest of these.
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