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Integers, Fractions and Decimals

Prime Factors, HCF and LCM

Pearson Edexcel International GCSE Mathematics A

Prime factors

  • A prime factor is a factor that is a prime number.
  • Answer form: a product of powers of its prime factors, with × between the primes.
Method: writing a number as a product of prime factors
  1. Split the number into two factors, one of them prime, in a factor tree or a ladder.
  2. Split again until every end is prime.
  3. Write all the primes multiplied together, smallest first, with a power for each repeated prime.
  • Write the primes multiplied. Never write them as a list separated by commas or added.
  • Every factor in the answer is prime. Never leave a composite or decimal factor in the product.
  • A number given with a power of 10: write 10n = 2n × 5n and collect the powers of 2 and of 5.

Products, multiples and powers of a prime factor form

Method: a product or power of numbers in prime factor form
  1. Write every number, including any integer multiplier, as a product of powers of primes.
  2. Combine the powers of each prime with the index laws.
  3. Write the answer smallest prime first, in the form asked for.
Worked example: product of numbers given in prime factor form

M = 2 × 32 × 53 and N = 23 × 5 × 72. Work out the value of M × N2. Give your answer as a product of powers of its prime factors.

N2 = 26 × 52 × 74

M × N2 = 2 × 32 × 53 × 26 × 52 × 74

27 × 32 × 55 × 74

HCF and LCM

Highest common factor (HCF)Lowest common multiple (LCM)
Primes usedonly the primes in every numberevery prime in any of the numbers
Power of each primethe lower powerthe higher power
  • For two numbers: LCM = product of the two numbers ÷ HCF.
  • An unknown power in a number given in prime factor form is fixed by a given HCF (the lower power) or LCM (the higher power).
Method: HCF and LCM of integers
  1. Write each number as a product of prime factors, in a factor tree or ladder.
  2. Place the primes in a Venn diagram: primes shared by both numbers in the overlap, the rest outside it.
  3. HCF: multiply the primes in the overlap.
  4. LCM: multiply every prime in the diagram.
  • Working for prime factors, HCF or LCM: the factor tree, ladder, Venn diagram or lists of factors or multiples.
  • Asked for "a product of powers of prime factors": give the HCF or LCM in that form, not as an integer.
Worked example: HCF and LCM of multiples of numbers in prime factor form

A = 22 × 33 × 7 and B = 24 × 3 × 52
(a) Write down the highest common factor (HCF) of 3A and 2B. Give your answer as a product of powers of its prime factors.
(b) Find the lowest common multiple (LCM) of 3A and 2B. Give your answer as a product of powers of its prime factors.

3A = 22 × 34 × 7 and 2B = 25 × 3 × 52

(a) 22 × 3

(b) 25 × 34 × 52 × 7

Trap: HCF or LCM of multiples

Asked for the HCF of 3A and 2B, never give the HCF of A and B. Write each multiple in prime factor form first, because the multiplier raises the power of its own prime and that changes which power is lower or higher.

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