Prime Suspects
Every number is built from primes. Learn to break any number into its prime factors, then use them to crack HCF and LCM the way grade 9s do: fast, with working, no guessing.
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Prime Suspects 🔍
Every whole number bigger than 1 is built from **prime numbers** multiplied together, and there is only **one** way to build each one. Primes are the building blocks of the whole number system. Master how to break a number into its primes and you unlock a fast, reliable method for **HCF** and **LCM** — the kind examiners reward with full marks because you show working, not lucky guesses.
What makes a prime 🧱
A **prime number** has **exactly two** factors: **1 and itself**. So its only divisors are 1 and the number. Two facts examiners love to test: • **1 is NOT prime** — it has only *one* factor (itself), not two. • **2 is the only even prime** — every other even number is divisible by 2 as well, so it has more than two factors. The primes start: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 ...
Spot the prime
Which of these is a prime number?
- 51
- 29
- 1
- 57
Factors and multiples 🔢
Keep these two straight — they point in opposite directions: • A **factor** of a number divides into it exactly. The factors of 12 are 1, 2, 3, 4, 6, 12. Factors are **at most** the number. • A **multiple** of a number is in its times table. The multiples of 12 are 12, 24, 36, 48 ... Multiples are **at least** the number. Rule of thumb: **factors go into it, multiples come out of it.**
Factor or multiple?
A number that divides exactly into 20 is a _____ of 20, for example 5. A number in the 20 times table, such as 60, is a _____ of 20. A number with exactly two factors, such as 5, is a _____ number.
Breaking it down 🌳
**Prime factorisation** means writing a number as a product of primes only. A **factor tree** gets you there: keep splitting until every branch ends on a prime. Take **60**: 60 = 6 x 10 = (2 x 3) x (2 x 5). Collect the primes: 2 x 2 x 3 x 5. Then tidy repeated primes into **index form**: **60 = 2² × 3 × 5**. Because there is only one prime factorisation of a number, you get the same answer whichever factor pair you start with.
Order the factor-tree method
An interactive activity.
Largest prime factor
An interactive activity.
Match each number to its prime factorisation
- 12
- 18
- 20
- 45
- 2² × 3
- 2 × 3²
- 2² × 5
- 3² × 5
HCF and LCM from primes 🤝
Once you have the prime factorisations, HCF and LCM fall out — no listing, no guessing. Take **24** and **36**: • 24 = 2³ × 3 • 36 = 2² × 3² **HCF** (Highest Common Factor) = the primes they **share**, each to the **lowest** power: 2² × 3 = **12**. **LCM** (Lowest Common Multiple) = **every** prime that appears, each to the **highest** power: 2³ × 3² = **72**. Memory hook: **HCF = common, lowest. LCM = all, highest.**
Find the HCF
An interactive activity.
Find the LCM
An interactive activity.
HCF or LCM?
A shop packs 48 red sweets and 36 blue sweets into identical bags, using every sweet with none left over. What is the greatest number of bags they can make?
- 12 bags
- 144 bags
- 84 bags
- 6 bags