1.2 Number properties
Number properties
Everything on number properties that can come up in the WJEC GCSE Mathematics and Numeracy, in Unit 2 and Unit 3. Each skill below has the method, the mistake that costs the most marks, a worked example, and free practice marked the way WJEC marks it.
Practise number properties free
A worked example
Prime factors · [2 marks]
Write 396 as a product of its prime factors. Give your answer in index form.
- B1All the prime factors: 2, 2, 3, 3, 11
- B12² × 3² × 11
The exam skills (7)
Prime factors
- Split the number into any two factors, then keep splitting (a factor tree).
- Stop when every branch ends in a prime: 2, 3, 5, 7, 11…
- Multiply the primes together and write repeats as powers.
Watch out: Including 1 as a prime. 1 is not prime.
HCF
- HCF: the biggest number that divides into both. List factors or use prime factors.
- LCM: the smallest number both divide into. List multiples until one matches.
- Check: HCF × LCM = the two numbers multiplied.
Watch out: Mixing them up. HCF is small (a factor), LCM is big (a multiple).
Powers and roots
- Squared means × itself once: 2² = 4. Cubed means three of it: 2³ = 8.
- Know squares to 15² and cubes 2³, 3³, 4³, 5³, 10³.
- A root undoes a power: √49 = 7, ∛27 = 3.
Watch out: Thinking 2³ is 6. It is 2 × 2 × 2 = 8.
HCF and LCM
- Write both numbers as products of primes.
- HCF: the primes they share, each to the LOWER power.
- LCM: every prime that appears, each to the HIGHER power.
Watch out: Using the higher powers for the HCF.
LCM in context
- Things that repeat together are an LCM question.
- Find the LCM of the two times.
- Add it on to the starting time.
Watch out: Giving the LCM in minutes when the question asks for the time of day.
HCF in context
- Cutting or sharing into the biggest equal groups is an HCF question.
- Find the HCF of the two amounts.
- Answer with units.
Watch out: Using the LCM. The pieces must fit into both lengths, so they are a factor.
Using prime factors
- A square number has every prime to an even power.
- Look at the prime factors and find the ones with odd powers.
- Multiply by exactly those primes once each.
Watch out: Squaring the whole number. The question asks for the smallest multiplier.
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