Mathematics 4024 · Syllabus tracker
Hcf Lcm
Prime factorisation, highest common factors, lowest common multiples and the problems that use them.
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Quick revision
Hcf Lcm on one sheet
For the night before. The full notes teach the topic; this is just what you must remember.
At a glance
Prime factors, HCF and LCM, the problems they solve, and using primes for roots.
1Prime factorisation
- Every whole number above 1 can be written as a product of primes in exactly one way.
- Use a factor tree or keep dividing by the smallest prime (2, 3, 5, 7, 11, …).
- Write the answer in index form: 360 = 23 × 32 × 5.
Repeated division
- Divide by 2 until you can't.
- Then by 3, then 5, then 7…
- Stop at 1; collect equal primes as powers.
Worked example
Write 360 as a product of its prime factors.
- 360 ÷ 2 = 180, ÷ 2 = 90, ÷ 2 = 45.
- 45 ÷ 3 = 15, ÷ 3 = 5, ÷ 5 = 1.
- Primes used: 2, 2, 2, 3, 3, 5.
23 × 32 × 5
2HCF and LCM
- HCF (highest common factor): the biggest number that divides into both.
- LCM (lowest common multiple): the smallest number both divide into.
- From prime factors: HCF = primes they share, lowest power of each. LCM = every prime, highest power of each.
- A Venn diagram of the prime factors works too: HCF = the overlap; LCM = everything in the diagram.
Formulas
Worked example
Find the HCF and LCM of 84 and 120.
- 84 = 22 × 3 × 7; 120 = 23 × 3 × 5.
- HCF: 22 × 3 = 12.
- LCM: 23 × 3 × 5 × 7 = 840.
HCF 12, LCM 840
3Word problems
- 'Together again', 'at the same time', 'smallest number of each' → LCM.
- 'Largest possible', 'cut into equal pieces', 'share into identical groups' → HCF.
- Algebraic HCF: HCF of the numbers × the lowest power of each shared letter. HCF of 12x2y and 18xy3 is 6xy.
Worked example
Two lighthouses flash every 12 s and every 20 s. They flash together at 9:00. When do they next flash together?
- Together again → LCM.
- 12 = 22 × 3, 20 = 22 × 5, LCM = 22 × 3 × 5 = 60.
60 s later, at 9:01
4Square roots, cube roots and the smallest n
- A square number has every prime to an even power; a cube has every power a multiple of 3.
- √: halve every power. ∛: divide every power by 3. √(22 × 32 × 72) = 2 × 3 × 7 = 42.
- Smallest number to multiply by to make a square: supply whatever makes every power even.
Worked example
Find the smallest whole number n such that 120n is a cube number.
- 120 = 23 × 3 × 5.
- For a cube every power must be a multiple of 3.
- Need 32 × 52 more.
n = 225
✗Common mistakes
| Don’t write | Write instead |
|---|---|
| LCM of 18 and 24 = 18 × 24 = 432 | That is a common multiple, but the lowest is 72 |
| Writing 2 × 2 × 2 × 3 × 3 × 5 when 'index form' is asked | Write 23 × 32 × 5 |
| Including 1 in a prime factorisation | 1 is not prime — leave it out |
| Taking the highest powers for the HCF | HCF uses the LOWEST powers of shared primes |
★In the exam
- Always check the HCF divides both numbers and both numbers divide the LCM.
- In word problems, say which one you are finding (HCF or LCM) before calculating.
- Show the prime factorisations — they earn method marks even if a multiplication slips.