Types of Number, Primes, HCF and LCM
The foundation topic of the whole syllabus. It looks easy, which is exactly why marks get dropped — the definitions have edge cases that are tested deliberately, and HCF and LCM get swapped more than any other pair of terms in IGCSE Maths.
1. Prime numbers
A prime number has exactly two factors: 1 and itself.
The primes up to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
The edge cases that are actually tested:
1 is NOT prime. It has only one factor. The definition requires exactly two.
2 IS prime — and it is the only even prime. Every other even number is divisible by 2.
0 is not prime.
Numbers that look prime but aren’t:
| Number | Why not |
|---|---|
| 51 | 3 × 17 |
| 57 | 3 × 19 |
| 91 | 7 × 13 |
| 25, 49, 81 | square numbers |
Odd does not mean prime. 9, 15, 21, 25 and 27 are all odd and none is prime — confusing the two was a recorded error. Test divisibility by 2, 3, 5, 7, 11… up to the square root.
A useful check for 3: if the digit sum is divisible by 3, so is the number. 51 → 5+1 = 6 → divisible by 3.
2. Factors and multiples
Factors divide into a number exactly. Multiples are what you get by multiplying it.
- Factors of 28: 1, 2, 4, 7, 14, 28
- Multiples of 28: 28, 56, 84, 112, …
Factors are finite; multiples are infinite. Confusing them means answering a completely different question.
Find factors in pairs, working upwards, so none is missed: for 28 — 1×28, 2×14, 4×7. Stop when the pair meets.
“A factor” and “the list of factors” are different instructions. If asked for a factor, one is enough; if asked to list them, missing any loses the mark.
3. Other number types
| Type | Meaning | Examples |
|---|---|---|
| Integer | whole number, positive or negative | −3, 0, 7 |
| Natural | positive whole numbers | 1, 2, 3 |
| Square | n × n | 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 |
| Cube | n × n × n | 1, 8, 27, 64, 125 |
| Rational | can be written as a fraction a/b | 0.5, −3, 2/7, 0.3̇ |
| Irrational | cannot be written as a fraction | √2, √3, π |
A cube number is n³, not n × 3. And 64 is both a square (8²) and a cube (4³).
Rational includes integers, terminating decimals and recurring decimals. 0 is rational (0/1), and so is any negative whole number.
√4 is not irrational. Only roots that don’t come out exactly are — √2, √3, √5 are irrational, but √4 = 2 and √9 = 3 are rational. This distinction was taught explicitly in lessons and is regularly examined.
Negative numbers:
“Less than −5” means −6, −7, −8… — further from zero. Thinking −3 and −4 qualify was a documented error: on a number line they are to the right of −5, so they are greater.
The smallest number in a list is the most negative one. In {−8, 2, 6}, the smallest is −8.
4. Prime factorisation
Write a number as a product of prime factors, usually in index form.
Method — a factor tree, or repeated division by primes:
360 → 2 × 180 → 2 × 2 × 90 → 2 × 2 × 2 × 45 → 2 × 2 × 2 × 3 × 15 → 2 × 2 × 2 × 3 × 3 × 5
360 = 2³ × 3² × 5
Check: 8 × 9 × 5 = 360 ✓
Keep dividing until every branch is prime. Stopping early — leaving a 4 or a 9 in the answer — is the standard error.
Give the answer in index form unless told otherwise, and multiply back to check.
5. HCF and LCM
HCF — Highest Common Factor: the largest number that divides into both. LCM — Lowest Common Multiple: the smallest number both divide into.
Using prime factorisation
360 = 2³ × 3² × 5 and 84 = 2² × 3 × 7
HCF — take the LOWEST power of each COMMON prime: 2² × 3 = 12
LCM — take the HIGHEST power of EVERY prime that appears: 2³ × 3² × 5 × 7 = 2520
HCF is small, LCM is large. If your HCF is bigger than either number, or your LCM is smaller, you have swapped them. This mix-up was the single most frequent error in the topic.
A check that always works: HCF × LCM = the product of the two numbers. Here: 12 × 2520 = 30 240, and 360 × 84 = 30 240 ✓
Which one does the question want?
This is the real skill, and the language is consistent:
| Use HCF when… | Use LCM when… |
|---|---|
| splitting into groups | events repeat and you want them to coincide |
| finding the largest possible size | finding the smallest or next time |
| ”greatest number of identical bags" | "ring every 4 and 6 minutes — when together?“ |
| cutting into equal pieces with none left | buying packs so nothing is left over |
“Every” is an LCM signal; “each group” or “greatest” points to HCF. Reading for these keywords is faster and more reliable than trying to reason it out fresh each time.
Example (LCM): one bell rings every 4 minutes, another every 6. They ring together at 12:00. When next?
- LCM(4, 6) = 12 → 12:12
Example (HCF): 24 pencils and 36 rubbers, split into identical packs with none left over. Largest number of packs?
- HCF(24, 36) = 12 packs
6. Squares and cubes from prime factors
A number is a perfect square when every prime factor has an even power. A number is a perfect cube when every power is a multiple of 3.
Example: what is the smallest number to multiply 360 by to make a square?
- 360 = 2³ × 3² × 5 — the powers of 2 and 5 are odd
- multiply by 2 × 5 = 10 → 3600 = 60² ✓
7. Mistakes that cost marks
Calling 1 a prime number.
Saying 2 isn’t prime because it’s even.
Assuming odd numbers are prime — 51, 57, 91 are not.
Confusing factors with multiples.
Missing factors by not working in pairs.
Swapping HCF and LCM — the biggest error in this topic.
Not finishing a factor tree.
Leaving prime factorisation out of index form.
Using the highest power for HCF or the lowest for LCM.
Misreading negative numbers, treating −3 as less than −5.
Calling √4 irrational.
Saying n³ means n × 3.
Frequently asked questions
Is 1 a prime number? No — a prime has exactly two factors, and 1 has one.
Is 2 a prime number? Yes — and it is the only even prime.
What is the difference between factors and multiples? Factors divide into a number (finite list); multiples are the times table of it (infinite).
What is the HCF? The largest number that divides into both — take the lowest power of each common prime.
What is the LCM? The smallest number both divide into — take the highest power of every prime present.
How do I know whether to use HCF or LCM? HCF for splitting into the largest equal groups; LCM for repeating events coinciding, or the smallest number that works.
How do I write a number as a product of primes? Use a factor tree, dividing by primes until every branch is prime, then write it in index form.
What is a rational number? Any number expressible as a fraction — including integers and recurring decimals.
Is √4 irrational? No — it equals 2. Only roots that don’t work out exactly, like √2, are irrational.
How do I check my HCF and LCM? HCF × LCM should equal the product of the two numbers.
Quick revision checklist
- I know the definition of a prime and can list primes to 50
- I know 1 is not prime and 2 is
- I don’t assume odd numbers are prime
- I can tell factors from multiples and find factors in pairs
- I know square, cube, rational and irrational numbers
- I know √4 is rational but √2 is not
- I can order negative numbers correctly
- I can write a number as a product of primes in index form
- I finish every branch of the factor tree
- I can find HCF (lowest powers, common primes)
- I can find LCM (highest powers, all primes)
- I know HCF is small and LCM is large
- I can decide from the wording which is needed
- I check with HCF × LCM = product of the numbers
- I can use the even-power test for square numbers
These notes cover types of number, primes, HCF and LCM in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus and formula list for your own exam series.
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