IGCSE Maths · Number · Core & Extended

Number Basics: Primes, Factors, Multiples, Rationals & Sets

This is where the whole IGCSE Number topic is built from — every later idea (indices, surds, standard form, percentages) rests on these foundations. The content below is shared between Core and Extended; anywhere the two tiers split, you’ll see a tag telling you which one it applies to. There’s a lot here on purpose — take it steadily, section by section, and use the challenge rounds to check things have actually stuck before moving on.

1. Natural Numbers and Integers — The Starting Point

Natural numbers are the counting numbers: 1, 2, 3, 4, 5, … (some definitions include 0 — for IGCSE, just be comfortable that these are the numbers you’d use to count objects).

Integers are a bigger family: all the whole numbers, positive, negative, and zero.

Integers = { …, −3, −2, −1, 0, 1, 2, 3, … }
−4 −3 −2 −1 0 1 2
Everyday example A lift in a building goes from Floor 3 (basement) to Floor 12. If you’re on Floor 2 and go down 5 floors, where do you end up?
2 − 5 = −3 (Basement Floor 3). Negative integers aren’t abstract — you use them for temperature, money owed, sea level, and lift floors all the time.
Worked example — ordering integers Arrange in order from smallest to largest: −8, 3, −1, 0, −5.
Think of the number line: further left = smaller.
−8, −5, −1, 0, 3
Try it — Warm-up
  1. Is 0 an integer? Is it a natural number?
  2. Write down all integers between −4 and 3 (not including −4 and 3).
  3. The temperature is −6°C. It rises by 9°C. What is the new temperature?
  4. Arrange in order, smallest first: −12, 4, −1, −9, 2.
  5. A submarine is at −120 m (120 m below sea level). It rises 45 m, then descends 30 m. What is its new position?
Show answers
  1. Yes it’s an integer; whether it counts as a natural number depends on the definition you’re using — for IGCSE, just know both views exist.
  2. −3, −2, −1, 0, 1, 2
  3. 3°C
  4. −12, −9, −1, 2, 4
  5. −120 + 45 − 30 = −105 m

2. Prime Numbers — The Building Blocks of Every Whole Number

A prime number has exactly two factors: 1 and itself. The number 1 is not prime (it only has one factor). Every other whole number greater than 1 is either prime or can be broken down into primes — this is why primes are called the “building blocks” of numbers.

First 8 primes: 2, 3, 5, 7, 11, 13, 17, 19

Notice 2 is the only even prime — every other even number can be divided by 2, so it has more than two factors.

Find them yourself — Sieve of Eratosthenes (do this in class) Write out the numbers 1 to 30. Cross out 1 (not prime). Circle 2, then cross out every multiple of 2. Circle the next number left (3), then cross out every multiple of 3. Keep going. Whatever is left uncrossed is prime. You should end up circling: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
Worked example — prime factorisation Express 72 as a product of its prime factors.
72 → 8 × 9 → (2 × 2 × 2) × (3 × 3)
So 72 = 2³ × 3²
Try it — Primes
  1. Is 51 a prime number? (Hint: check if 3 divides it.)
  2. Is 97 a prime number?
  3. List all prime numbers between 20 and 40.
  4. Express 60 as a product of its prime factors.
  5. Express 196 as a product of its prime factors, giving your answer in index form.
  6. Express 500 as a product of its prime factors.
Show answers
  1. No — 51 = 3 × 17, so it has more than two factors.
  2. Yes — it isn’t divisible by 2, 3, 5, or 7 (the primes up to √97), so 97 is prime.
  3. 23, 29, 31, 37
  4. 60 = 2² × 3 × 5
  5. 196 = 2² × 7²
  6. 500 = 2² × 5³

3. Square and Cube Numbers, Square Roots and Cube Roots

A square number is what you get when you multiply a whole number by itself. A cube number is what you get when you multiply a whole number by itself three times. Going backwards — finding what number was squared or cubed — gives you the square root (√) and cube root (³√).

n123456789101112131415
n²149162536496481100121144169196225
n1234510
n³1827641251000

Memorise both tables completely — the exam expects instant recall, without a calculator. You’ll rely on this heavily when we get to square roots, cube roots, and later, surds.

Worked example Write down the value of √169 and ³√125.
√169 = 13 (since 13² = 169)    ³√125 = 5 (since 5³ = 125)
Try it — Squares, Cubes & Roots
  1. What is 13²?
  2. What is 6³? (Not in the memory table — work it out.)
  3. Find √225.
  4. Find ³√64.
  5. Which number between 1 and 100 is both a square number and a cube number?
Show answers
  1. 169
  2. 216 (6 × 6 × 6)
  3. 15
  4. 4
  5. 64 (it’s 8² and also 4³)

4. Common Factors and Common Multiples — HCF and LCM

A factor of a number divides into it exactly. A multiple of a number is what you get by multiplying it by a whole number. When two numbers share factors or multiples, we call them common factors/multiples.

  • HCF (Highest Common Factor) — the biggest number that divides into both numbers exactly.
  • LCM (Lowest Common Multiple) — the smallest number that both numbers divide into exactly.
Method 1 — listing (good for smaller numbers) Find the HCF of 18 and 24.
Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Common factors: 1, 2, 3, 6 → HCF = 6
Method 2 — prime factors (better for larger numbers) Find the HCF and LCM of 24 and 36.
24 = 2³ × 3
36 = 2² × 3²
HCF: take the lowest power of each common prime → 2² × 3 = 12
LCM: take the highest power of each prime that appears → 2³ × 3² = 72
Everyday example Two buses leave the same stop at the same time. Bus A returns every 15 minutes, Bus B returns every 20 minutes. After how many minutes will they next leave together?
This is asking for the LCM of 15 and 20.
15 = 3 × 5,   20 = 2² × 5  →  LCM = 2² × 3 × 5 = 60 minutes
Try it — HCF & LCM
  1. List the factors of 18.
  2. Find the HCF of 16 and 40.
  3. Find the LCM of 8 and 12.
  4. Find the HCF and LCM of 90 and 120 using prime factorisation.
  5. Find the HCF and LCM of 42 and 56.
  6. Challenge: Two joggers run laps of a track. One takes 6 minutes per lap, the other 8 minutes. If they start together, after how many minutes will they both be back at the start at the same time — and how many laps will each have run?
Show answers
  1. 1, 2, 3, 6, 9, 18
  2. 8
  3. 24
  4. 90 = 2 × 3² × 5, 120 = 2³ × 3 × 5 → HCF = 2 × 3 × 5 = 30, LCM = 2³ × 3² × 5 = 360
  5. 42 = 2 × 3 × 7, 56 = 2³ × 7 → HCF = 2 × 7 = 14, LCM = 2³ × 3 × 7 = 168
  6. LCM of 6 and 8 = 24 minutes. Jogger 1 runs 4 laps, Jogger 2 runs 3 laps.

5. Rational and Irrational Numbers

A rational number can be written as a fraction a/b, where a and b are integers (b ≠ 0). This includes whole numbers, fractions, and decimals that terminate or repeat.

An irrational number cannot be written as an exact fraction — its decimal goes on forever without repeating. The most famous examples are π and square roots of numbers that aren’t perfect squares, like √2 or √7.

Sorting exercise — think before you check Which of these are rational, and which are irrational?
0.5,   √9,   √10,   π,   7,   0.333…

Rational: 0.5 (= 1/2), √9 (= 3, a whole number), 7, 0.333… (= 1/3)
Irrational: √10 (10 isn’t a perfect square), π

Keep this in mind — this is exactly the idea behind surds, which we’ll build on very soon.

Try it — Rational or Irrational?
  1. Is √49 rational or irrational? Explain why.
  2. Is √50 rational or irrational?
  3. Is 0.6 rational? Write it as a fraction in simplest form.
  4. Is √81 − 5 rational or irrational?
  5. Is 2π rational or irrational?
Show answers
  1. Rational — √49 = 7, a whole number.
  2. Irrational — 50 is not a perfect square.
  3. Rational — 0.6 = 3/5.
  4. Rational — √81 = 9, so √81 − 5 = 4, a whole number.
  5. Irrational — any non-zero multiple of an irrational number is still irrational.

6. Reciprocals

The reciprocal of a number is 1 divided by that number — flip it upside down. Multiplying a number by its reciprocal always gives 1.

Reciprocal of a/b is b/a
Worked examples Reciprocal of 5 is 1/5
Reciprocal of 3/4 is 4/3
Reciprocal of 0.2 (= 1/5) is 5
Reciprocal of −4 is −1/4 (the sign stays the same)
Try it — Reciprocals
  1. What is the reciprocal of 8?
  2. What is the reciprocal of 2/9?
  3. What is the reciprocal of 1.25?
  4. What is the reciprocal of −3/5?
  5. A number’s reciprocal is 1/12. What is the number?
Show answers
  1. 1/8
  2. 9/2
  3. 1.25 = 5/4, so the reciprocal is 4/5
  4. −5/3
  5. 12

7. Ordering Numbers

Before you can compare quantities in context — prices, temperatures, test scores — you need to be fluent with the symbols mathematicians use to say “bigger than,” “smaller than,” or “equal to.”

= — is equal to
≠ — is not equal to
> — is greater than
< — is less than
⩾ — is greater than or equal to
⩽ — is less than or equal to
Worked example — ordering mixed numbers Put these in order, smallest first: 0.6, 3/5, 0.55, 2/3.
Convert everything to decimals to compare fairly: 0.6, 0.6, 0.55, 0.667
Order: 0.55, 3/5, 0.6, 2/3 (3/5 and 0.6 are equal, so either order between them is acceptable, but list 3/5 first here to show working clearly).
Try it — Ordering
  1. Insert >, < or = between: −3 ___ −7
  2. Insert >, < or = between: 2/5 ___ 0.4
  3. Order from smallest to largest: 0.7, 5/8, 0.65, 3/4.
  4. Write down all integer values of x that satisfy −2 ⩽ x < 3.
Show answers
  1. −3 > −7
  2. 2/5 = 0.4
  3. 5/8 (0.625), 0.65, 0.7, 3/4 (0.75)
  4. −2, −1, 0, 1, 2 — five values.

8. The Four Operations — Negatives, Fractions and Decimals

You’ve used +, −, × and ÷ since primary school — the IGCSE twist is doing them confidently with negative numbers, fractions, decimals, and mixed numbers together, always respecting the correct order of operations (BODMAS/BIDMAS).

Worked example — negative numbers Work out −8 − (−3) + (−5).
Subtracting a negative is the same as adding: −8 + 3 − 5 = −10
Worked example — order of operations Work out 5 + 3 × (7 − 4)².
Brackets first: (7 − 4) = 3. Then the power: 3² = 9. Then multiply: 3 × 9 = 27. Then add: 5 + 27 = 32
Worked example — fractions and mixed numbers Work out 2¾ − 1⅓.
Convert to improper fractions: 11/4 − 4/3. Common denominator 12: 33/12 − 16/12 = 17/12 = 1 5/12
Try it — Four Operations
  1. Work out −6 + (−4) − (−10).
  2. Work out 4 − 2 × 3 + 5.
  3. Work out (6 − 9)² × 2.
  4. Work out 1½ + 2⅔.
  5. Work out 3.5 × 4 − 2.25.
Show answers
  1. 0
  2. 3
  3. 18
  4. 4 1/6
  5. 11.75

9. Sets and Venn Diagrams

A set is just a well-defined collection of things — numbers, letters, objects, anything. Sets are one of the most useful tools in maths because they let us describe “groups that overlap” clearly, using both notation and pictures (Venn diagrams).

Core notation Core & Extended

n(A) — the number of elements in set A
A′ — the complement of A (everything NOT in A)
ℰ — the universal set (everything under consideration)
A ∪ B — union: everything in A OR B (or both)
A ∩ B — intersection: everything in BOTH A AND B
{x : …} — set-builder notation, e.g. {x : x is a natural number}
Reading a two-set Venn diagram ℰ A B 2, 4 6 3, 9 1, 5, 7 Here, A = {2, 4, 6}, B = {3, 6, 9}, and ℰ = {1, 2, 3, 4, 5, 6, 7, 9}.
A ∩ B = {6} (the overlap)    A ∪ B = {2, 3, 4, 6, 9}    n(A) = 3    A′ = {1, 3, 5, 7, 9}
Try it — Sets (Core & Extended)
  1. ℰ = {1, 2, 3, …, 10}. P = {even numbers}, Q = {multiples of 3}. List the elements of P and Q.
  2. Using the sets in Q1, find P ∩ Q.
  3. Using the sets in Q1, find P ∪ Q.
  4. Find n(P).
  5. Find P′.
Show answers
  1. P = {2, 4, 6, 8, 10}, Q = {3, 6, 9}
  2. P ∩ Q = {6}
  3. P ∪ Q = {2, 3, 4, 6, 8, 9, 10}
  4. n(P) = 5
  5. P′ = {1, 3, 5, 7, 9}

Extra notation and three-set Venn diagrams Extended only

Extended students need a few more symbols, and need to be comfortable reading Venn diagrams with three overlapping sets (Core only ever uses two).

∈ — “is an element of”   e.g. 3 ∈ A
∉ — “is not an element of”   e.g. 5 ∉ A
∅ — the empty set (a set with nothing in it)
A ⊆ B — A is a subset of B (every element of A is also in B)
A ⊈ B — A is not a subset of B
Worked example — element notation and subsets ℰ = {1, 2, 3, 4, 5, 6}, A = {2, 4, 6}, B = {2, 4}.
Is 4 ∈ A? Yes. Is 5 ∈ A? No, so 5 ∉ A.
Is B ⊆ A? Check: every element of B (2, 4) is also in A. Yes, B ⊆ A.
Worked example — three-set Venn diagram ℰ = {1, 2, …, 12}. A = {multiples of 2}, B = {multiples of 3}, C = {multiples of 4}.
A = {2, 4, 6, 8, 10, 12}, B = {3, 6, 9, 12}, C = {4, 8, 12}.
Notice C ⊆ A (every multiple of 4 is also a multiple of 2) — this is exactly the kind of relationship a 3-set Venn diagram is built to show clearly. The region where all three circles overlap is A ∩ B ∩ C = {12}.
Try it — Extended Sets Notation
  1. ℰ = {1, 2, 3, …, 9}, A = {1, 3, 5, 7, 9}. Is 6 ∈ A or 6 ∉ A?
  2. Let A = {2, 4, 6} and B = {2, 4, 6, 8, 10}. Is A ⊆ B? Explain.
  3. Let A = {1, 2} and B = {3, 4}. What is A ∩ B?
  4. ℰ = {1, 2, …, 20}. A = {factors of 20}, B = {multiples of 5}, C = {even numbers}. List A, B, C, then find A ∩ B ∩ C.
Show answers
  1. 6 ∉ A
  2. Yes — every element of A (2, 4, 6) is also in B.
  3. A ∩ B = ∅ (the empty set — no elements in common)
  4. A = {1, 2, 4, 5, 10, 20}, B = {5, 10, 15, 20}, C = {2, 4, 6, …, 20} → A ∩ B ∩ C = {10, 20}

10. Quick-Fire Challenge Round

A fast-paced round to finish on — no long working, just instinct and quick checking. Good for a competitive last few minutes.

True or False? (answer instantly, then check)
  1. Every prime number is odd.
  2. 0 is a multiple of every number.
  3. The HCF of two numbers is never bigger than the smaller number.
  4. √2 × √2 is rational.
  5. The reciprocal of a number greater than 1 is always less than 1.
  6. ∅ ⊆ A is true for every set A.
Show answers
  1. False — 2 is prime and even.
  2. True — 0 = n × 0 for any n.
  3. True — a common factor can never exceed the smaller number.
  4. True — √2 × √2 = 2, a whole number.
  5. True — e.g. reciprocal of 4 is 1/4.
  6. True — the empty set is a subset of every set, by definition. Ext.
Spot the Error Each “solution” below has exactly one mistake. Find it and correct it.
  1. “36 = 2² × 3² × 2, so 36 = 2³ × 3².” (check the arithmetic first)
  2. “The HCF of 12 and 18 is 36.” (HCF or LCM — which one is this really?)
  3. “√25 is irrational because it has a square root sign in it.”
  4. “The reciprocal of 5 is −5.”
Show answers
  1. 36 = 2² × 3² (four factors of 2 and 3 multiplied gives 4×9=36, not 8×9). Correct: 36 = 2² × 3².
  2. 36 is actually the LCM of 12 and 18. The HCF is 6.
  3. Wrong — √25 = 5, a whole number, so it’s rational. Having a root sign doesn’t automatically make something irrational; it depends whether the number inside is a perfect square.
  4. The reciprocal of 5 is 1/5, not −5. (−5 would be the negative, which is a completely different operation from “reciprocal.”)
Number riddle I am a two-digit number. I am a prime number. Reverse my digits and I’m still prime. The sum of my digits is 8. What number am I?
Reveal answer17 (reversed: 71, also prime; 1 + 7 = 8) — 71 also works as a valid reversed pair, but 17 is the intended “smaller” answer; either is acceptable if a student finds 71 first and justifies it.

11. Mixed Practice — Bringing It All Together

Core & Extended
  1. Express 84 as a product of its prime factors.
  2. Find the HCF and LCM of 45 and 75.
  3. Is √121 rational or irrational? Justify your answer.
  4. Find the reciprocal of 3/7.
  5. ℰ = {1, 2, 3, …, 15}. A = {multiples of 3}, B = {multiples of 5}. Find A ∩ B and A ∪ B.
  6. Work out (2 − 8) × 3 + 5².
  7. Order from smallest to largest: 7/10, 0.72, 0.7, 3/4.
Show answers
  1. 84 = 2² × 3 × 7
  2. 45 = 3² × 5, 75 = 3 × 5² → HCF = 3 × 5 = 15, LCM = 3² × 5² = 225
  3. Rational — √121 = 11, a whole number.
  4. 7/3
  5. A = {3, 6, 9, 12, 15}, B = {5, 10, 15} → A ∩ B = {15}, A ∪ B = {3, 5, 6, 9, 10, 12, 15}
  6. (2 − 8) × 3 + 25 = (−6 × 3) + 25 = −18 + 25 = 7
  7. 0.7, 7/10 (=0.7 — equal, list together), 0.72, 3/4 (=0.75) → 0.7, 7/10, 0.72, 3/4
Extended-level mixed set Extended only
  1. Let A = {x : x is a prime number less than 20}. List the elements of A.
  2. ℰ = {1, 2, …, 30}. A = {multiples of 6}, B = {multiples of 9}. Is A ⊆ B? Find A ∩ B.
  3. State whether true or false: every square number is rational.
  4. Find two numbers whose HCF is 1 (they are called “co-prime”) between 14 and 22.
Show answers
  1. A = {2, 3, 5, 7, 11, 13, 17, 19}
  2. A = {6, 12, 18, 24, 30}, B = {9, 18, 27} → A ⊈ B (e.g. 6 ∈ A but 6 ∉ B). A ∩ B = {18}
  3. True — every square number is a whole number, and every whole number can be written as a fraction over 1.
  4. Many valid answers, e.g. 15 and 22 (HCF = 1)

Practice at Home

This is a large set — you don’t need to rush. Work through it steadily; it’s designed to give you enough repetition that every idea in this note becomes automatic.

Part A — Natural numbers & integers
  1. Arrange in order, smallest first: −15, 8, −2, 0, −20, 6.
  2. A lift starts at Floor 4, goes down 9 floors, then up 3 floors. Where does it end up?
  3. Write down all integers between −7 and 2.
Part B — Primes and prime factorisation
  1. List all prime numbers between 40 and 60.
  2. Express 132 as a product of its prime factors.
  3. Express 180 as a product of its prime factors.
  4. Is 91 a prime number? Show your working.
Part C — Squares, cubes and roots
  1. Find √196.
  2. Find ³√1000.
  3. Work out 14² and 15² without a calculator.
  4. Find a square number between 100 and 200.
Part D — HCF and LCM
  1. Find the HCF and LCM of 132 and 180 (use your answers from Part B).
  2. Find the HCF and LCM of 28 and 42.
  3. Three bells ring every 12, 18, and 24 minutes. If they all ring together at 9:00, when will they next all ring together?
Part E — Rational, irrational and reciprocals
  1. Sort into rational and irrational: √64, √11, 0.454545…, π, 5/9, √200.
  2. Find the reciprocal of 0.75 and of 6.
  3. Find the reciprocal of −2/7.
  4. Is 3 + √4 rational or irrational? Explain.
Part F — Ordering numbers
  1. Insert >, < or = between: −9 ___ −4
  2. Order from smallest to largest: 5/6, 0.8, 0.83, 4/5.
  3. List all integers x satisfying −5 < x ⩽ 1.
Part G — The four operations
  1. Work out −12 + (−5) − (−20).
  2. Work out 6 + 4 × (9 − 6)².
  3. Work out 3⅓ − 1¾.
  4. Work out 5.6 ÷ 0.8 − 1.5.
Part H — Sets and Venn diagrams (Core & Extended)
  1. ℰ = {1, 2, …, 20}. A = {factors of 20}, B = {prime numbers}. List A and B, then find A ∩ B, A ∪ B, and n(A ∪ B).
  2. ℰ = {1, 2, …, 12}. C = {even numbers}, D = {factors of 12}. Draw and label a two-set Venn diagram showing C and D, with every element of ℰ placed correctly.
Part I — Extended only Extended
  1. ℰ = {1, 2, …, 25}. A = {multiples of 5}, B = {multiples of 10}. Is B ⊆ A? Explain your answer using the definition of a subset.
  2. ℰ = {whole numbers 1 to 30}. A = {multiples of 2}, B = {multiples of 3}, C = {multiples of 5}. Find A ∩ B ∩ C.
  3. Give an example of two sets A and B (using numbers 1–10) where A ∩ B = ∅.

Key Vocabulary Recap

Natural number — a counting number (1, 2, 3, …)
Integer — a whole number, positive, negative, or zero
Prime number — exactly two factors: 1 and itself
HCF — Highest Common Factor
LCM — Lowest Common Multiple
Rational number — can be written as a fraction
Irrational number — decimal never terminates or repeats
Reciprocal — 1 divided by the number; flip a fraction
BODMAS/BIDMAS — order of operations: Brackets, Powers, Divide/Multiply, Add/Subtract
Universal set (ℰ) — everything under consideration
Subset (⊆) — every element of one set is in another Ext.

Syllabus Reference

This note covers Cambridge IGCSE Mathematics (0580) sections C1.1, C1.2, C1.5 and C1.6 (Core subject content) and E1.1, E1.2, E1.5 and E1.6 (Extended subject content) — Types of number, Sets, Ordering, and The four operations. The two tiers overlap almost completely; the only differences are the element/subset notation (∈, ∉, ∅, ⊆, ⊈) and three-set Venn diagrams, both marked “Extended only” above.

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