Prime Factors, HCF & LCMEdexcel GCSE Maths: Revision notes
Section 1
What are prime numbers, factors and multiples?
A prime number has exactly two factors: 1 and itself (e.g. 2, 3, 5, 7, 11, 13...). Note that 1 is not prime, and 2 is the only even prime number.
A factor (or divisor) of a number divides into it exactly with no remainder. A multiple of a number is any number in its times table.
- Factors of 12: 1, 2, 3, 4, 6, 12
- Multiples of 12: 12, 24, 36, 48...
Common factors are factors shared by two or more numbers; common multiples are multiples shared by two or more numbers.
Section 2
How do we find the prime factorisation of a number?
Prime factorisation expresses a number as a product of its prime factors, usually in index notation. The unique factorisation theorem states every integer greater than 1 has exactly one prime factorisation (ignoring order).
Method: use a factor tree — repeatedly split the number into factors until every branch ends in a prime.
- 60 = 2 × 30 = 2 × 2 × 15 = 2 × 2 × 3 × 5 = 2² × 3 × 5
Always write the final answer in index form (e.g. 2² × 3 × 5, not 2 × 2 × 3 × 5) — this is the expected exam format.
Section 3
How do we find the HCF using prime factors?
The highest common factor (HCF) of two or more numbers is the largest number that divides into all of them.
Method using prime factors: write each number's prime factorisation, then multiply together the lowest power of each common prime factor.
- 60 = 2² × 3 × 5, and 84 = 2² × 3 × 7
- Common primes: 2 and 3, lowest powers: 2² and 3¹
- HCF = 2² × 3 = 12
Section 4
How do we find the LCM using prime factors?
The lowest common multiple (LCM) of two or more numbers is the smallest number that is a multiple of all of them.
Method using prime factors: write each number's prime factorisation, then multiply together the highest power of every prime factor that appears in any of the numbers.
- 60 = 2² × 3 × 5, and 84 = 2² × 3 × 7
- All primes involved: 2, 3, 5, 7, highest powers: 2², 3¹, 5¹, 7¹
- LCM = 2² × 3 × 5 × 7 = 420
A Venn diagram can also be used: place shared prime factors in the overlap, unique factors in the outer regions; HCF = product of the overlap, LCM = product of everything in the diagram.
Venn diagram for 12 (2²×3) and 18 (2×3²): overlap = 2×3 = 6 (this is the HCF); all regions multiplied = 2²×3² = 36 (this is the LCM)
Must Know
- Prime numbers have exactly two factors: 1 and themselves; 1 is not prime
- Prime factorisation writes a number as a product of primes in index form, using a factor tree
- HCF = product of the lowest powers of common prime factors
- LCM = product of the highest powers of all prime factors involved
- A Venn diagram of prime factors: the overlap gives the HCF, the whole diagram multiplied gives the LCM
- Every integer greater than 1 has a unique prime factorisation
That's the notes covered.
Carry on to the next subtopic.