Cambridge IGCSE 0580 (2028–2030)

Chapter 1: Review of Number Concepts

Student: Sabrina  |  Coursebook: Cambridge IGCSE Mathematics Core & Extended (3rd Ed)

πŸ’‘ What This Guide Covers

This is your complete, detailed study guide for Chapter 1. Every topic has:

  • Detailed explanations β€” each concept explained step by step with examples
  • Worked examples β€” see exactly how to solve each type of question
  • QR codes β€” scan to watch video tutorials
  • Practice questions β€” test yourself with answers included
πŸ“‹ Your Progress This Week
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πŸ”’ Types of Number

Syllabus: C1.1 / E1.1  |  Time needed: ~2 hours

Numbers are the building blocks of mathematics. In this topic, you'll learn to identify, classify, and work with different types of numbers. Understanding these types is essential because IGCSE questions often ask you to "write down a prime number" or "list all the square numbers" β€” you need to know exactly what each type means.

Natural Numbers

Natural numbers are the counting numbers β€” the first numbers you ever learned as a child. They start at 1 and go on forever:

Natural Numbers
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, … (and so on forever)

Key facts about natural numbers:

  • They start at 1 (not 0)
  • They are always positive (no negatives)
  • They are always whole numbers (no decimals or fractions)
  • They go on forever β€” there is no biggest natural number
πŸ’‘ Example

Question: Write down the first five natural numbers.

Answer: 1, 2, 3, 4, 5

Notice: we start at 1, not 0. Natural numbers are for counting β€” you can't count "zero apples" in real life!

⚠️ Common Mistake

Some people think 0 is a natural number. In IGCSE, natural numbers start at 1. If the question says "whole numbers" then 0 is included, but "natural numbers" means 1, 2, 3, …

Integers

Integers are all whole numbers β€” including negative numbers, zero, and positive numbers. Think of them as natural numbers plus their negatives and zero.

Integers
…, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …

Key facts about integers:

  • They include negative numbers: -1, -2, -3, …
  • They include zero: 0
  • They include positive numbers: 1, 2, 3, …
  • They are always whole numbers (no decimals or fractions like 2.5 or 3/4)
  • They go on forever in both directions
Worked Example: Is this an integer?
1
5 β€” Is this an integer? YES βœ… (it's a positive whole number)
2
-3 β€” Is this an integer? YES βœ… (it's a negative whole number)
3
0 β€” Is this an integer? YES βœ… (zero is an integer)
4
2.5 β€” Is this an integer? NO ❌ (it has a decimal part)
5
-7.1 β€” Is this an integer? NO ❌ (it has a decimal part)
6
3/4 β€” Is this an integer? NO ❌ (it's a fraction, not a whole number)
πŸ’‘ Memory Tip

Think of integers as all the numbers on a number line that land exactly on a mark β€” no in-between values. If you can point to it on a number line without it being between two marks, it's an integer.

Prime Numbers

A prime number is a number that has exactly two factors: 1 and itself. This means it can only be divided exactly by 1 and by itself β€” nothing else.

Prime Numbers β€” Definition
A prime number has EXACTLY two factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31

Why is 1 NOT a prime number?

  • A prime number must have exactly two factors
  • 1 only has one factor (1 itself)
  • Therefore, 1 is NOT prime

Why is 2 the only even prime?

  • Every even number (except 2) can be divided by 2
  • So every even number (except 2) has at least three factors: 1, 2, and itself
  • 2 only has two factors: 1 and 2 β€” so it IS prime

All prime numbers up to 31 (MEMORISE THESE):

Primes to Memorise
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31
Worked Example: Is 17 prime?
1
Find all the factors of 17: What numbers divide exactly into 17?
2
17 Γ· 1 = 17 βœ… (1 is a factor)
3
17 ÷ 2 = 8.5 ❌ (not a whole number)
4
17 ÷ 3 = 5.67 ❌ (not a whole number)
5
Keep going… none work until 17 Γ· 17 = 1 βœ…
6
Factors of 17: only 1 and 17 β€” that's exactly two factors!
βœ… Answer:
Yes, 17 is prime because it only has two factors: 1 and 17.
Worked Example: Is 15 prime?
1
Find all factors of 15: 1, 3, 5, 15
2
That's four factors, not two!
3
15 = 3 Γ— 5, so it has factors besides 1 and itself
βœ… Answer:
No, 15 is NOT prime because it has more than two factors (1, 3, 5, 15).
🚨 Common Mistakes

1 is NOT prime β€” it only has one factor

2 IS prime β€” it's the only even prime

0 is NOT prime β€” it has infinitely many factors

Negative numbers are NEVER prime β€” primes must be positive

Square Numbers

A square number is the result of multiplying a number by itself. We write this as nΒ² (read as "n squared").

Square Numbers
nΒ² = n Γ— n 1Β² = 1 2Β² = 4 3Β² = 9 4Β² = 16 5Β² = 25 6Β² = 36 7Β² = 49 8Β² = 64 9Β² = 81 10Β² = 100 11Β² = 121 12Β² = 144 13Β² = 169 14Β² = 196 15Β² = 225

Why are they called "square" numbers?

If you arrange dots in a square shape, square numbers are the ones that form a perfect square:

  • 4 dots can make a 2Γ—2 square
  • 9 dots can make a 3Γ—3 square
  • 16 dots can make a 4Γ—4 square
Worked Example: Calculate 12Β²
1
12Β² means 12 Γ— 12
2
12 Γ— 12 = 144
βœ… Answer:
12Β² = 144
⚠️ Common Mistake

12Β² β‰  24 β€” Don't multiply 12 Γ— 2! Squaring means multiplying the number by ITSELF: 12 Γ— 12 = 144, not 12 Γ— 2 = 24.

Cube Numbers

A cube number is the result of multiplying a number by itself three times. We write this as nΒ³ (read as "n cubed").

Cube Numbers
nΒ³ = n Γ— n Γ— n 1Β³ = 1 2Β³ = 8 3Β³ = 27 4Β³ = 64 5Β³ = 125 6Β³ = 216 7Β³ = 343 8Β³ = 512 9Β³ = 729 10Β³ = 1000
Worked Example: Calculate 4Β³
1
4Β³ means 4 Γ— 4 Γ— 4
2
4 Γ— 4 = 16
3
16 Γ— 4 = 64
βœ… Answer:
4Β³ = 64

Factors

A factor of a number is a number that divides into it exactly (with no remainder). Factors always come in pairs.

Factors
A factor divides a number exactly. Example: Factors of 12 = 1, 2, 3, 4, 6, 12 Because: 1Γ—12=12, 2Γ—6=12, 3Γ—4=12

How to find all factors of a number:

  1. Start with 1 and the number itself (they're always factors)
  2. Try 2, 3, 4, 5, … up to the square root of the number
  3. If n divides exactly, then both n and (number Γ· n) are factors
Worked Example: Find all factors of 36
1
1 Γ— 36 = 36 β†’ factors: 1 and 36
2
2 Γ— 18 = 36 β†’ factors: 2 and 18
3
3 Γ— 12 = 36 β†’ factors: 3 and 12
4
4 Γ— 9 = 36 β†’ factors: 4 and 9
5
6 Γ— 6 = 36 β†’ factor: 6 (just one, since both are the same)
6
Next would be 7, but 7 Γ— 5 = 35 (too small) and 7 Γ— 6 = 42 (too big) β€” stop!
βœ… Answer:
Factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36

Multiples

A multiple of a number is what you get when you multiply it by 1, 2, 3, 4, … (its times table). Multiples go on forever.

Multiples
Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, … Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, … Just keep adding the number to get the next multiple!

Key difference between factors and multiples:

  • Factors are smaller than (or equal to) the number β€” they divide INTO it
  • Multiples are larger than (or equal to) the number β€” the number divides INTO them
πŸ’‘ Memory Tip

Factors = "What can divide into this number?" (go DOWN)

Multiples = "What can this number divide into?" (go UP β€” the times table)

Prime Factor Decomposition (Product of Prime Factors)

Every whole number (greater than 1) can be broken down into a multiplication of prime numbers. This is called prime factorisation or product of prime factors.

Two methods to find prime factors:

  1. Factor tree β€” keep breaking the number into two factors until all branches end in primes
  2. Division method β€” keep dividing by the smallest prime that works
Method 1: Factor Tree β€” Express 72 as product of primes
1
72 = 8 Γ— 9 (break into two factors)
2
8 = 2 Γ— 4 (break 8 further)
3
4 = 2 Γ— 2 (break 4 β€” now we have primes!)
4
9 = 3 Γ— 3 (break 9 β€” now we have primes!)
5
All the prime ends: 2, 2, 2, 3, 3
βœ… Answer:
72 = 2 Γ— 2 Γ— 2 Γ— 3 Γ— 3 = 2Β³ Γ— 3Β²
Method 2: Division Method β€” Express 180 as product of primes
1
180 Γ· 2 = 90
2
90 Γ· 2 = 45
3
45 Γ· 3 = 15
4
15 Γ· 3 = 5
5
5 Γ· 5 = 1 (done!)
βœ… Answer:
180 = 2 Γ— 2 Γ— 3 Γ— 3 Γ— 5 = 2Β² Γ— 3Β² Γ— 5

HCF (Highest Common Factor) and LCM (Lowest Common Multiple)

HCF β€” Highest Common Factor

The HCF of two or more numbers is the largest number that divides into all of them exactly. It's also called the Greatest Common Divisor (GCD).

Worked Example: Find HCF of 24 and 36
1
Write both as products of primes:
24 = 2 Γ— 2 Γ— 2 Γ— 3 = 2Β³ Γ— 3
36 = 2 Γ— 2 Γ— 3 Γ— 3 = 2Β² Γ— 3Β²
2
Find the common primes with the lowest power:
Common 2s: min(3,2) = 2Β²
Common 3s: min(1,2) = 3ΒΉ
3
Multiply: 2Β² Γ— 3 = 4 Γ— 3 = 12
βœ… Answer:
HCF(24, 36) = 12

LCM β€” Lowest Common Multiple

The LCM of two or more numbers is the smallest number that all of them divide into exactly.

Worked Example: Find LCM of 24 and 36
1
Write both as products of primes:
24 = 2Β³ Γ— 3
36 = 2Β² Γ— 3Β²
2
Find the highest power of each prime that appears:
Highest 2s: max(3,2) = 2Β³
Highest 3s: max(1,2) = 3Β²
3
Multiply: 2Β³ Γ— 3Β² = 8 Γ— 9 = 72
βœ… Answer:
LCM(24, 36) = 72
πŸ’‘ Quick Check

HCF check: Does 12 divide into 24? Yes (24Γ·12=2). Does 12 divide into 36? Yes (36Γ·12=3). βœ…

LCM check: Does 24 divide into 72? Yes (72Γ·24=3). Does 36 divide into 72? Yes (72Γ·36=2). βœ…

▢️ Watch & Learn β€” Prime Numbers & Factors
β–Ά
Prime Numbers
Corbettmaths
β–Ό
β–Ά
Factors
Corbettmaths
β–Ό
β–Ά
LCM & HCF using Product of Primes
Corbettmaths
β–Ό
✏️ Practice Questions β€” Try These!
1. List all prime numbers between 20 and 40.
23, 29, 31, 37
2. Express 180 as a product of prime factors.
180 = 2 Γ— 90 = 2 Γ— 2 Γ— 45 = 2 Γ— 2 Γ— 5 Γ— 9 = 2Β² Γ— 3Β² Γ— 5
3. Find the HCF and LCM of 24 and 36.
HCF = 12 (common primes: 2 Γ— 2 Γ— 3)
LCM = 72 (highest powers: 2Β³ Γ— 3Β²)
4. Write down all square numbers from 1 to 225.
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225 (1Β² to 15Β²)
5. Is 1 a prime number? Explain why or why not.
No. A prime number must have exactly two factors (1 and itself). 1 only has one factor, so it is NOT prime.
6. Find all factors of 48.
1, 2, 3, 4, 6, 8, 12, 16, 24, 48
7. Which of these are integers: 5, -3, 0, 2.5, -7.1, 3/4?
5, -3, 0 are integers. 2.5, -7.1, and 3/4 are NOT integers (they have decimal/fraction parts).
▢️ More Practice Videos
β–Ά
Primes, Factors & Multiples β€” Practice Questions
Corbettmaths
β–Ό
β–Ά
Factors, Multiples and Primes
Maths Genie
β–Ό

πŸ“Š Sets & Venn Diagrams

Syllabus: C1.2 / E1.2  |  Time needed: ~1.5 hours

A set is a collection of objects or numbers. We use special notation and diagrams to describe and compare sets. This topic is about learning the "language" of sets β€” the symbols and diagrams mathematicians use.

Set Notation β€” What Each Symbol Means

Set notation is like a special shorthand. Once you learn these symbols, you can read and write mathematical statements very efficiently.

SymbolNameWhat It MeansExample
{ }Curly bracketsMakes a set β€” lists the elements inside{1, 2, 3} β€” the set containing 1, 2, and 3
∈Element of"Is in the set"3 ∈ {1, 2, 3} β€” "3 is in the set"
βˆͺUnion"OR" β€” everything in either set (or both)A βˆͺ B β€” all elements in A or B or both
∩Intersection"AND" β€” only what's in both setsA ∩ B β€” elements in both A and B
A'Complement"NOT in A" β€” everything outside AA' β€” everything in the universal set that is NOT in A
n(A)Number in setHow many elements are in An(A) = 3 means set A has 3 elements
βˆ…Empty setA set with nothing in itβˆ… or { } β€” no elements at all
UUniversal setEverything we're talking aboutU = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
πŸ’‘ How to Remember Union vs Intersection

βˆͺ (Union) looks like a U β€” think "either Usually" = OR = everything

∩ (Intersection) looks like an n β€” think "and" = AND = only the overlap

Venn Diagrams

A Venn diagram uses overlapping circles inside a rectangle to visually show how sets relate to each other. The rectangle represents the universal set (everything), and each circle represents one set.

πŸ“ How to Draw a Venn Diagram
  1. Draw a rectangle β€” this is the universal set (U)
  2. Draw overlapping circles inside β€” one for each set
  3. Write numbers in the overlap if they're in BOTH sets
  4. Write numbers in the non-overlapping part if they're in only ONE set
  5. Write numbers outside the circles if they're in U but not in any set
Worked Example: Draw a Venn Diagram for A = {1, 2, 3, 4} and B = {3, 4, 5, 6}
1
Find the overlap: What numbers are in BOTH A and B? β†’ 3 and 4 go in the middle
2
Circle A only: What's in A but NOT in B? β†’ 1 and 2 go in the left part of circle A
3
Circle B only: What's in B but NOT in A? β†’ 5 and 6 go in the right part of circle B
4
A βˆͺ B (union): Everything in either circle β†’ {1, 2, 3, 4, 5, 6}
5
A ∩ B (intersection): Only the overlap β†’ {3, 4}
βœ… Answer:
A βˆͺ B = {1, 2, 3, 4, 5, 6} and A ∩ B = {3, 4}
Worked Example: Complement of a Set
1
Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {2, 4, 6, 8, 10}
2
A' means "everything NOT in A"
3
Go through U and remove anything in A: remove 2, 4, 6, 8, 10
4
What's left: 1, 3, 5, 7, 9
βœ… Answer:
A' = {1, 3, 5, 7, 9}
▢️ Watch & Learn β€” Sets & Venn Diagrams
β–Ά
Set Notation and Venn Diagrams Explained
GCSE Maths Revision
β–Ό
β–Ά
Venn Diagrams β€” Revision Session
Corbettmaths
β–Ό
✏️ Practice Questions
1. P = {2, 4, 6, 8, 10} and Q = {3, 6, 9, 12}. Find P βˆͺ Q and P ∩ Q.
P βˆͺ Q = {2, 3, 4, 6, 8, 9, 10, 12}
P ∩ Q = {6} (only 6 is in both)
2. In a class of 30 students, 18 play football and 12 play cricket. 5 play both. How many play neither?
Football only: 18 - 5 = 13. Cricket only: 12 - 5 = 7. Total playing: 13 + 5 + 7 = 25. Neither: 30 - 25 = 5
3. If U = {1,2,3,4,5,6,7,8,9,10} and A = {2,4,6,8,10}, what is A'?
A' = {1, 3, 5, 7, 9} β€” these are the numbers NOT in A.

⬆️ Powers and Roots

Syllabus: C1.3 / E1.3  |  Time needed: ~2 hours

Powers (indices) and roots are opposite operations. A power means multiply a number by itself repeatedly. A root asks: "what number, multiplied by itself, gives this answer?"

Squares and Cubes β€” Tables You MUST Memorise

Square numbers are made by multiplying a number by itself once: nΒ² = n Γ— n

n123456789101112131415
nΒ²149162536496481100121144169196225

Cube numbers are made by multiplying a number by itself twice: nΒ³ = n Γ— n Γ— n

n12345678910
nΒ³1827641252163435127291000

Square Roots (√)

The square root is the opposite of squaring. If n² = x, then √x = n. It asks: "what number, when multiplied by itself, gives this?"

Worked Example: What is √144?
1
Ask yourself: what number squared = 144?
2
From the squares table: 12Β² = 144
βœ… Answer:
√144 = 12

Cube Roots (βˆ›)

The cube root is the opposite of cubing. If nΒ³ = x, then βˆ›x = n.

Worked Example: What is βˆ›27?
1
Ask yourself: what number cubed = 27?
2
From the cubes table: 3Β³ = 27
βœ… Answer:
βˆ›27 = 3
⚠️ Common Mistakes

2Β³ β‰  2 Γ— 3 β€” 2Β³ means 2 Γ— 2 Γ— 2 = 8, NOT 6!

3Β² β‰  3 Γ— 2 β€” 3Β² means 3 Γ— 3 = 9, NOT 6!

√16 β‰  8 β€” √16 = 4 (because 4Β² = 16)

Worked Example: Calculate 5² + √81 - ³√27
1
5Β² = 5 Γ— 5 = 25
2
√81 = ? β†’ 9Β² = 81, so √81 = 9
3
³√27 = ? β†’ 3Β³ = 27, so ³√27 = 3
4
25 + 9 - 3 = 31
βœ… Answer:
5² + √81 - ³√27 = 25 + 9 - 3 = 31
▢️ Watch & Learn β€” Powers & Roots
β–Ά
What are Indices? β€” IGCSE 0580
Mr Oh Math
β–Ό
✏️ Practice Questions
1. Calculate: (a) 7² (b) 4³ (c) √144 (d) ³√64
(a) 49 (b) 64 (c) 12 (d) 4
2. Calculate: 3Β² + 4Β²
3Β² + 4Β² = 9 + 16 = 25 (note: this is NOT the same as (3+4)Β² = 49!)
3. Find the value of: √169 + ³√125 - 2³
13 + 5 - 8 = 10