What is Number Theory in the context of Cambridge IGCSE Mathematics?

Number Theory is a branch of mathematics that deals with the properties and relationships of numbers, particularly integers. In the Cambridge IGCSE Mathematics curriculum, it includes topics such as prime numbers, factors, multiples, and divisibility rules. Understanding these concepts is essential for solving problems related to numbers and their properties.

How do you determine if a number is prime in Number Theory?

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. To determine if a number is prime, you can check if it is divisible by any prime numbers up to its square root. If it is not divisible by any of these, then it is a prime number. This concept is crucial in Number Theory and is covered in the Cambridge IGCSE Mathematics syllabus.

What are the divisibility rules covered in Cambridge IGCSE Number Theory?

Divisibility rules help determine if one number is divisible by another without performing division. Common rules include: a number is divisible by 2 if it is even, by 3 if the sum of its digits is divisible by 3, by 5 if it ends in 0 or 5, and by 10 if it ends in 0. These rules are fundamental in Number Theory and are part of the Cambridge IGCSE Mathematics curriculum.

What is the significance of the greatest common divisor (GCD) in Number Theory?

The greatest common divisor (GCD) of two or more integers is the largest positive integer that divides each of the numbers without leaving a remainder. In Number Theory, the GCD is important for simplifying fractions and solving problems involving ratios. The concept is included in the Cambridge IGCSE Mathematics syllabus, where students learn to calculate the GCD using methods such as prime factorization and the Euclidean algorithm.

How are prime factorization and least common multiple (LCM) related in Number Theory?

Prime factorization involves expressing a number as a product of its prime factors. The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both. In Number Theory, prime factorization is used to find the LCM by taking the highest power of all prime factors present in the numbers. This relationship is an important concept in the Cambridge IGCSE Mathematics curriculum, helping students solve problems involving multiples and divisors.

Why is "Including $1$ in a list of prime numbers" a common mistake in Number Theory?

Why it happens: Students remember the rule "primes have factors 1 and themselves" but forget that this requires **exactly two** distinct factors — and 1 has only one. How to avoid it: Memorise the formal definition: a prime is an integer **greater than 1** with exactly two positive factors. Cross-check by listing factors. Source: 0580/22 Jun 2022 — examiner report

Why is "Marking a negative number as prime" a common mistake in Number Theory?

Why it happens: When a question asks for primes from a mixed list (with negatives), students miss the requirement that primes are positive. How to avoid it: Read the definition every time: primes are **positive integers greater than 1**. If it's negative, immediately exclude it. Source: 0580/12 Oct/Nov 2023 — examiner report

Why is "Swapping the HCF and LCM rules in prime-factorisation questions" a common mistake in Number Theory?

Why it happens: The rules sound similar — "common primes" for both — and under exam pressure students apply the wrong power. How to avoid it: Anchor it: **HCF = lower power**, **LCM = larger power**. The HCF must be small (it has to fit inside both numbers), so the powers must be small. Source: 0580/42 May/Jun 2024 — examiner report Q2

Why is "Leaving prime factorisation as a long product instead of in index form" a common mistake in Number Theory?

Why it happens: Students stop as soon as they get to primes (2 × 2 × 2 × 3 × 3 × 5) and forget the question asked for index form. How to avoid it: Always read the final part of the question. If it says "in index form", group the repeats: 2^3 × 3^2 × 5. Source: 0580/12 Oct/Nov 2022 — examiner report Q4

Why is "Writing an irrational answer as a rounded decimal" a common mistake in Number Theory?

Why it happens: Students reach for the calculator and write something like √(17) 4.123, treating it as a final answer. How to avoid it: If the question requires the *exact* irrational form, leave it as a surd. A rounded decimal turns the answer rational and loses the accuracy mark. Source: 0580/42 Feb/Mar 2023 — examiner report

Why is "Calling a recurring decimal irrational" a common mistake in Number Theory?

Why it happens: Students think "never-ending" implies irrational, but they confuse *recurring* (repeats) with *irrational* (non-repeating non-terminating). How to avoid it: Rule: every recurring decimal can be written as a fraction, so every recurring decimal is rational. 0.3 = 1/3.

Why is "Flipping a mixed number directly without converting first" a common mistake in Number Theory?

Why it happens: The reciprocal of 21/4 is wrongly written as 1/24/1 or 1/214 left unsimplified. How to avoid it: Always convert mixed to improper first: 21/4 = 9/4, then flip to get 4/9.

Why is "Calling $0$ a natural number" a common mistake in Number Theory?

Why it happens: Different textbooks use different conventions; some include 0 in the naturals. How to avoid it: Cambridge IGCSE 0580 follows the convention that **natural numbers start at 1**. 0 is an integer but not natural. Source: Cambridge IGCSE 0580 syllabus 2025-2027 — Section 1.1

Why is "Writing $\sqrt{16}$ as irrational" a common mistake in Number Theory?

Why it happens: Surds *look* irrational, so students assume any expression with √(\ ) is irrational without simplifying. How to avoid it: Always simplify first. √(16) = 4 is an integer, hence rational. Only √(n) where n is **not** a perfect square is irrational. Source: 0580/22 May/Jun 2023 — examiner report

Cambridge IGCSE Mathematics (0580)

Number Theory

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Short Notes - Number Theory

Detailed Study Notes

Detailed notes on Number for Cambridge IGCSE Mathematics, covering key concepts, explanations, examples, and exam-focused revision points.

Number Theory Study Notes — Cambridge IGCSE 0580 Maths Extended (2026)

Number Theory underpins the rest of the Number topic. Get the four families of numbers, primes, squares and cubes, and reciprocals exam-ready — with the precise vocabulary Cambridge 0580 mark schemes credit.

At a glance

Numbers are organised into nested families: $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$ .
Rational = can be written as $\dfrac{p}{q}$ with integer $p, q$ and $q \ne 0$ . Includes terminating AND recurring decimals.
Irrational = decimal that goes on forever without repeating. e.g. $\pi$ , $\sqrt{2}$ , $\sqrt{3}$ .
Prime = exactly TWO factors (1 and itself). $1$ is not prime; $2$ IS prime.
Square numbers: $1, 4, 9, 16, 25, \ldots$ . Cube numbers: $1, 8, 27, 64, 125, \ldots$ . Memorise to $15^{2}$ and $5^{3}$ .
Reciprocal of $a$ (where $a \ne 0$ ) is $\dfrac{1}{a}$ . Multiplying any number by its reciprocal gives 1.
Mark schemes credit precise vocabulary: 'integer', 'rational', 'irrational', 'reciprocal'.

What you’ll learn

Mapped to the Cambridge IGCSE 0580 syllabus (2025-2027).

E1.1 — Identify and use natural numbers, integers (positive, negative and zero), prime numbers, square numbers, cube numbers, common factors, common multiples, rational and irrational numbers, real numbers and reciprocals.

The four families of numbers

Every number you meet on a 0580 paper sits in one of these nested families. Knowing which family it belongs to tells you what you can do with it.

Numbers in the IGCSE syllabus are organised into four nested families. Each family is a SUBSET of the next.

Natural numbers ( $\mathbb{N}$ ) are the positive whole numbers used for counting: $1, 2, 3, 4, 5, \ldots$ . Some conventions include $0$ ; Cambridge usually doesn't.

Integers ( $\mathbb{Z}$ ) are all whole numbers — positive, negative and zero: $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$ . Every natural number is an integer.

Rational numbers ( $\mathbb{Q}$ ) are numbers that can be written as $\dfrac{p}{q}$ where $p$ and $q$ are integers and $q \ne 0$ . Examples:

$\tfrac{3}{4}$ is rational ( $p = 3, q = 4$ ).
$-7$ is rational ( $p = -7, q = 1$ ).
$0.625$ is rational ( $p = 5, q = 8$ ).
$0.\overline{3}$ is rational ( $p = 1, q = 3$ ).

A decimal is rational when it either terminates (e.g. $0.5$ , $0.625$ ) or recurs (e.g. $0.\overline{3} = 0.333\ldots$ , $0.\overline{142857}$ ).

Real numbers ( $\mathbb{R}$ ) are every point on the number line. Real numbers split into rationals and irrationals.

Irrational numbers are real numbers that are NOT rational — their decimal expansion goes on forever WITHOUT settling into a repeating pattern. The classics:

$\pi \approx 3.14159265\ldots$
$\sqrt{2} \approx 1.41421356\ldots$
$\sqrt{3}, \sqrt{5}, \sqrt{7}, \ldots$ (square root of any non-perfect square is irrational).
$e \approx 2.71828\ldots$ (you'll meet this in A-level, not IGCSE).

Each family is a subset of the next; irrational numbers fill the real-number region outside the rationals.

$\mathbb{N}$ ⊂ $\mathbb{Z}$ ⊂ $\mathbb{Q}$ ⊂ $\mathbb{R}$ — each family contains the previous one.
Rational = fraction of two integers. Includes terminating AND recurring decimals.
Irrational = neither terminates nor recurs. $\pi$ and $\sqrt{2}$ are the textbook examples.
Real numbers cover every point on the number line.

Prime numbers — recognition

Primes are integers with exactly two factors. The full machinery for using them (factor trees, HCF, LCM) lives in the Factors and Multiples notes.

A prime number is a positive integer with exactly two distinct factors: $1$ and itself.

The first ten primes are $2, 3, 5, 7, 11, 13, 17, 19, 23, 29$ .

A prime has exactly two factors; a positive integer above 1 with more than two is composite.

A few facts you should know on sight:

$2$ is the only even prime. Every other even number is divisible by $2$ as well as by $1$ and itself, so has at least three factors.
$1$ is NOT prime. It only has ONE factor (itself).
$0$ is NOT prime either — it has infinitely many factors.

A positive integer above $1$ that is NOT prime is called a composite number. So $4, 6, 8, 9, 10, 12, \ldots$ are composite.

You'll meet primes again — the heavy lifting (factor trees, HCF, LCM) belongs to the Factors and Multiples notes. For Number Theory, you just need to be able to recognise primes and know the language.

Prime = exactly TWO distinct factors.
$2$ is the only even prime.
$1$ is not prime; $0$ is not prime.
Composite = positive integer above 1 that is not prime.

Square and cube numbers

Recognising squares and cubes on sight saves serious time on the calculator paper.

A square number is the product of an integer with itself. The first fifteen are $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225.$

A cube number is an integer multiplied by itself three times. The first five are $1, 8, 27, 64, 125.$

A square number forms a square; a cube number forms a cube.

Square and cube numbers come up everywhere — they're hidden inside surd-simplification questions, geometric mean questions, and ratio problems with areas/volumes. Memorise the lists above; they will pay back the time you invested.

(How to USE squares and cubes — square roots, cube roots, fractional indices, surd manipulation — lives in the Exponents and Surds notes.)

Memorise squares to $15^{2} = 225$ .
Memorise cubes to $5^{3} = 125$ .
Square numbers come up inside surd simplification.
Cube numbers come up inside volume problems.

Reciprocals

The reciprocal of a number is what you multiply by to get $1$ . Useful in fraction division, gradient questions, and ratio.

The reciprocal of a non-zero number $a$ is $\dfrac{1}{a}$ .

The defining property: $a \times \dfrac{1}{a} = 1$ for any $a \ne 0$ .

Examples:

Reciprocal of $5$ is $\dfrac{1}{5}$ .
Reciprocal of $\dfrac{2}{3}$ is $\dfrac{3}{2}$ (flip the fraction).
Reciprocal of $0.4$ : rewrite as $\dfrac{2}{5}$ , flip to $\dfrac{5}{2} = 2.5$ .

To find a reciprocal, swap the numerator and the denominator.

Reciprocals are involved in:

Dividing fractions — divide by a fraction = multiply by its reciprocal: $\dfrac{1}{2} \div \dfrac{3}{4} = \dfrac{1}{2} \times \dfrac{4}{3} = \dfrac{2}{3}$ .
Negative gradients of perpendicular lines — see Coordinate Geometry. Two perpendicular gradients multiply to give $-1$ , so each is the negative reciprocal of the other.
Inverse proportion — if $y$ varies inversely as $x$ , then $y$ is proportional to the reciprocal of $x$ .

Zero has no reciprocal — division by zero is undefined.

Reciprocal of $a$ is $\dfrac{1}{a}$ (for $a \ne 0$ ).
Flip the fraction: reciprocal of $\dfrac{p}{q}$ is $\dfrac{q}{p}$ .
Multiplying any number by its reciprocal gives $1$ .
Zero has no reciprocal.

Ordering and the number line

Compare numbers using <, >, ≤, ≥. Negative numbers and decimals are where most slips happen.

On a number line, larger numbers sit to the right. The standard inequality symbols:

Symbol	Meaning
$<$	strictly less than
$\le$	less than or equal to
$>$	strictly greater than
$\ge$	greater than or equal to

Negative-number ordering. $-3$ is LESS than $-1$ , even though $3 > 1$ . Closer to zero = larger when both are negative. So $-3 < -1$ .

Numbers increase to the right — so for negatives, the one closer to zero is larger.

Decimal ordering. Compare digit by digit from the left. $0.4 < 0.45 < 0.5$ . (Pad with zeros if useful: $0.4 = 0.40, 0.5 = 0.50$ .)

Mixed forms. Convert to the same form before ordering. $\tfrac{2}{3}$ vs $0.65$ ? $\tfrac{2}{3} = 0.\overline{6} \approx 0.667$ , so $\tfrac{2}{3} > 0.65$ .

On a number line, larger numbers are to the RIGHT.
For negatives: closer to zero = larger.
Compare decimals digit by digit from the left.
Convert mixed forms (fraction/decimal/percentage) to one form before comparing.

Quick recap

Four families: $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$ .
Rational = fraction of integers; irrational = non-terminating, non-recurring decimal.
Prime = exactly two factors. $2$ IS prime; $1$ is NOT.
Squares to $15^{2}$ and cubes to $5^{3}$ should be on the tip of your tongue.
Reciprocal of $a$ is $\dfrac{1}{a}$ ; flip the fraction; zero has no reciprocal.
Larger numbers sit to the right on the number line; for negatives, closer to zero is larger.

Memorise this

Verbatim phrases and definitions Cambridge mark schemes credit.

Integer — a whole number, positive, negative or zero.
Rational number — a number that can be written as $\dfrac{p}{q}$ where $p, q$ are integers and $q \ne 0$ .
Irrational number — a real number that cannot be written as a fraction of two integers.
Prime number — a positive integer with exactly two distinct factors: 1 and itself.
Reciprocal — the multiplicative inverse: the reciprocal of $a$ is $\dfrac{1}{a}$ .

How it’s examined

Number Theory itself rarely carries more than 1-2 marks per question on its own — it's the foundation that other topics test. Expect to identify whether a given number is rational/irrational, name the smallest prime, write the reciprocal, or order a list of mixed fractions/decimals/percentages. The vocabulary appears on every Paper 2; mark schemes credit the keywords ("irrational", "reciprocal", "integer") even on follow-on questions.

Sources: Cambridge IGCSE Mathematics 0580 syllabus 2025-2027 (E1.1); 0580/22 May/Jun 2024 — Q1 (identifying number families); 0580/42 Oct/Nov 2024 — Q2 (rational vs irrational); 0580 Examiner Reports 2022-2024. Last reviewed 2026-05-05.

Take this whole topic with you

Step-by-step worked examples — Number Theory

Step-by-step solutions to past-paper-style questions on number theory, written exactly the way a tutor would explain them at the board.

Question type:

Question patterns to master — Number Theory

Almost every number theory exam question is one of these shapes. Learn to spot each one and you will always know how to start.

Identify & classify

Recognise it by

The question hands you a list or set of numbers and asks you to write down, list or state which ones are a certain type — prime, integer, irrational, square, and so on.

How to approach it

Evaluate every expression first ( $\sqrt{16}=4$ , $1.\overline{3}=\frac{4}{3}$ ), then test each value against the definition — not its appearance. Tick or cross each candidate in turn.

Common trap

Classifying by how a number looks. $\sqrt{16}$ looks irrational but equals $4$ ; $0$ is an integer but not a natural number; $1$ is not prime. Examiner reports flag these slips every series.

Direct calculation

Recognise it by

A single instruction — find, work out, write … as a product of prime factors — on one number or one pair. One method, one answer.

How to approach it

Pick the standard method (factor tree for prime factorisation; prime factorisation for HCF and LCM) and show every line of working — method marks are awarded even if the final value slips. Give the answer in exactly the form asked for.

Common trap

Leaving prime factorisation as a long product instead of index form, or swapping the HCF and LCM rules. The mark scheme only credits the requested form.

Word problem

Recognise it by

A real-world context — tiles, buses, lights, ribbons — with no formula stated. You have to choose the method yourself.

How to approach it

Strip the context back to the maths. Largest size that fits into / divides both points to HCF. Next time things happen together / smallest common amount points to LCM. Then solve it as a direct calculation.

Common trap

The word 'largest' lures candidates into LCM and 'together' into HCF — it is usually the opposite. Decide from what the quantity must do, never from the adjective.

Multi-step problem

Recognise it by

You are given several pieces — for example an HCF, an LCM and one of the numbers — and asked for the missing one. The pieces must be chained together.

How to approach it

Spot the identity that links them — $a \times b = \text{HCF}(a,b) \times \text{LCM}(a,b)$ — substitute the known values, solve, then verify by prime-factorising your answer.

Common trap

Falling back on trial-and-error instead of using the identity, then running out of time — or not checking the final answer.

Show that / prove

Recognise it by

The words show that, prove, explain why or justify. The answer is already given — your job is to produce the reasoning.

How to approach it

Argue the general case algebraically wherever you can — write the numbers in terms of $n$ — then finish with an explicit sentence stating that the required result has been reached.

Common trap

Testing one or two examples instead of proving the general case, and omitting the concluding line. Examiner reports stress that a 'show that' with no final statement loses the last mark.

1Classify a list of numbers (rational, irrational, integer, prime)
ExtendedIdentify & classify• Adapted from 0580/22 May/Jun 2023 Q1• types of numbers, rational, irrational, prime
Question
From the list $\{ -7,\ \dfrac{2}{3},\ \sqrt{16},\ \sqrt{17},\ 0,\ 13,\ 1.\overline{3} \}$ write down (a) all integers, (b) all prime numbers, (c) all irrational numbers.
Step-by-step solution
1. Step 1
  Evaluate any expressions first. $\sqrt{16} = 4$ (an integer). $\sqrt{17}$ does not simplify to a rational, so it stays as it is. $1.\overline{3} = \dfrac{4}{3}$ — a recurring decimal is rational.
2. Step 2
  Integers are whole numbers, including zero and negatives. From the list: $-7,\ \sqrt{16}=4,\ 0,\ 13$ .
3. Step 3
  Prime numbers are integers greater than 1 with exactly two factors. Check $4$ (factors $1,2,4$ — not prime) and $13$ (factors $1, 13$ only — prime). The number $0$ and the negatives are excluded by definition.
4. Step 4
  Irrational numbers cannot be written as a fraction of two integers. $\sqrt{17}$ is irrational because $17$ is not a perfect square; everything else in the list is rational.
Answer
(a) $-7,\ 4,\ 0,\ 13$ (b) $13$ (c) $\sqrt{17}$
Examiner tip
Examiners commonly award zero marks if students forget that $0$ is an integer or if they list a negative number as prime. State $\sqrt{17}$ in surd form — converting to a rounded decimal can lose the irrational mark.
2Write a number as a product of prime factors
CoreDirect calculation• Adapted from 0580/12 Oct/Nov 2022 Q4• prime factorisation, index notation
Question
Write $360$ as a product of its prime factors, giving your answer in index form.
Step-by-step solution
1. Step 1
  Use a factor tree. Divide $360$ by the smallest prime that goes in: $360 \div 2 = 180$ .
2. Step 2
  Continue: $180 \div 2 = 90$ , $90 \div 2 = 45$ .
3. Step 3
  $45$ is odd, so move to the next prime: $45 \div 3 = 15$ , $15 \div 3 = 5$ .
4. Step 4
  $5$ is itself prime, so stop. Collect the primes: $360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5$ .
  $360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5$
5. Step 5
  Write in index form by grouping repeats.
  $360 = 2^{3} \times 3^{2} \times 5$
Answer
$360 = 2^{3} \times 3^{2} \times 5$
Examiner tip
The 0580 mark scheme awards full credit only when the answer is given in index form. Students who write $2 \times 2 \times 2 \times 3 \times 3 \times 5$ frequently lose the final accuracy mark.
3Find the HCF and LCM using prime factorisation
ExtendedDirect calculation• Adapted from 0580/42 May/Jun 2024 Q2(b)• HCF, LCM, prime factorisation
Question
Given $A = 2^{4} \times 3^{2} \times 5$ and $B = 2^{2} \times 3^{3} \times 7$ , find (a) the highest common factor (HCF) of $A$ and $B$ , and (b) the lowest common multiple (LCM) of $A$ and $B$ .
Step-by-step solution
1. Step 1
  For the HCF, take each prime that appears in both numbers, raised to the lower power.
2. Step 2
  Common primes: $2$ (lower power $2^{2}$ ) and $3$ (lower power $3^{2}$ ). The primes $5$ and $7$ are not in both, so they are excluded.
  $\text{HCF} = 2^{2} \times 3^{2} = 4 \times 9 = 36$
3. Step 3
  For the LCM, take every prime that appears in either number, raised to the higher power.
4. Step 4
  Primes used: $2^{4}$ , $3^{3}$ , $5$ , $7$ .
  $\text{LCM} = 2^{4} \times 3^{3} \times 5 \times 7 = 16 \times 27 \times 35 = 15120$
Answer
(a) $\text{HCF} = 36$ (b) $\text{LCM} = 15{,}120$
Examiner tip
The 2024 examiner report notes that many candidates confuse the HCF and LCM rules. Memorise: HCF = lower power of common primes; LCM = higher power of all primes.
4Determine whether an expression is rational or irrational
ExtendedShow that / prove• Adapted from 0580/42 Feb/Mar 2023 Q15• rational, irrational, surds
Question
Without using a calculator, state whether each value is rational or irrational. Show working.\n\n(a) $(\sqrt{5} + \sqrt{20})^{2}$ (b) $\sqrt{8} \times \sqrt{2}$ (c) $\dfrac{\pi}{2}$
Step-by-step solution
1. Step 1
  (a) Simplify $\sqrt{20} = 2\sqrt{5}$ , so the expression becomes $(\sqrt{5} + 2\sqrt{5})^{2} = (3\sqrt{5})^{2}$ .
2. Step 2
  Square it: $(3\sqrt{5})^{2} = 9 \times 5 = 45$ — an integer, so rational.
  $(\sqrt{5} + \sqrt{20})^{2} = 45$
3. Step 3
  (b) $\sqrt{8} \times \sqrt{2} = \sqrt{16} = 4$ — rational.
4. Step 4
  (c) $\pi$ is irrational, and dividing by a non-zero rational ( $2$ ) cannot make it rational. So $\dfrac{\pi}{2}$ is irrational.
Answer
(a) Rational ( $=45$ ) (b) Rational ( $=4$ ) (c) Irrational
Examiner tip
Examiners reward candidates who simplify the surds first and then state the conclusion. A bare answer of "rational" with no working scores zero in the 2023 mark scheme.
5Use a divisibility test to identify factors
CoreDirect calculation• divisibility, factors
Question
Decide whether $4{,}356$ is divisible by (a) $3$ , (b) $4$ , (c) $9$ . Justify each answer.
Step-by-step solution
1. Step 1
  (a) Sum the digits: $4 + 3 + 5 + 6 = 18$ . Since $18$ is divisible by $3$ , $4{,}356$ is divisible by $3$ .
2. Step 2
  (b) Look at the last two digits: $56$ . Since $56 \div 4 = 14$ exactly, $4{,}356$ is divisible by $4$ .
3. Step 3
  (c) The digit sum is $18$ , which is divisible by $9$ , so $4{,}356$ is divisible by $9$ .
Answer
Yes to all three: divisible by $3$ , $4$ and $9$ .
Examiner tip
Stating the rule ("digit sum divisible by 3") earns the method mark even if you make an arithmetic slip. Skip the rule and you lose both marks.
6Identify square, cube and triangular numbers from a sequence
CoreIdentify & classify• Adapted from 0580/22 Oct/Nov 2023 Q5• square numbers, cube numbers, triangular numbers
Question
From the set $\{ 1, 4, 8, 10, 16, 27, 36, 64 \}$ write down (a) the square numbers, (b) the cube numbers, (c) the numbers that are both square and cube.
Step-by-step solution
1. Step 1
  Square numbers: $n^{2}$ for some integer $n$ . Test each: $1=1^{2}$ , $4=2^{2}$ , $16=4^{2}$ , $36=6^{2}$ , $64=8^{2}$ .
2. Step 2
  Cube numbers: $n^{3}$ for some integer $n$ . From the list: $1=1^{3}$ , $8=2^{3}$ , $27=3^{3}$ , $64=4^{3}$ .
3. Step 3
  Both square and cube means $n^{6}$ . From the list: $1=1^{6}$ and $64=2^{6}$ .
Answer
(a) $1, 4, 16, 36, 64$ (b) $1, 8, 27, 64$ (c) $1$ and $64$
Examiner tip
Many candidates forget that $1$ qualifies as a square, cube and triangular number simultaneously. Always include $1$ unless the question explicitly excludes it.
7Find the reciprocal of a mixed expression
CoreDirect calculation• reciprocal, rational numbers
Question
Find the reciprocal of $2\frac{1}{4}$ , giving your answer as a fraction in lowest terms.
Step-by-step solution
1. Step 1
  Convert to an improper fraction: $2\frac{1}{4} = \dfrac{9}{4}$ .
2. Step 2
  The reciprocal of $\dfrac{a}{b}$ is $\dfrac{b}{a}$ , so the reciprocal of $\dfrac{9}{4}$ is $\dfrac{4}{9}$ .
3. Step 3
  $\dfrac{4}{9}$ is already in lowest terms (since $\gcd(4,9)=1$ ).
Answer
$\dfrac{4}{9}$
Examiner tip
Always convert mixed numbers to improper fractions before flipping. Flipping the parts separately ( $\dfrac{1}{2}\frac{4}{1}$ ) is a common error.
8List all factors of an integer and identify the primes
ExtendedDirect calculation• Adapted from 0580/12 May/Jun 2023 Q3• factors, primes, listing
Question
Write down all the factors of $72$ . From your list, state (a) the prime factors and (b) the total number of factors.
Step-by-step solution
1. Step 1
  Pair the factors systematically starting from $1$ : $1 \times 72$ , $2 \times 36$ , $3 \times 24$ , $4 \times 18$ , $6 \times 12$ , $8 \times 9$ . Stop when the factor pairs meet in the middle.
2. Step 2
  Collect the pairs in order: $1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72$ .
3. Step 3
  Pick the primes (integers $>1$ with exactly two factors): $2$ and $3$ .
4. Step 4
  Count the list to get $12$ factors. Cross-check with $\tau(72)$ : since $72 = 2^{3} \times 3^{2}$ , $\tau = (3+1)(2+1) = 12$ . ✓
Answer
Factors: $1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72$ . (a) Prime factors: $2, 3$ . (b) $12$ factors.
Examiner tip
Missing one factor pair (commonly $8 \times 9$ or $4 \times 18$ ) is the most frequent slip on this question. Pair systematically — stop only when the pairs meet — then count.
9HCF word problem — the largest tile that fits without cutting
ExtendedWord problem• Adapted from 0580/22 Oct/Nov 2023 Q11• HCF, word problem, context
Question
A rectangular floor measures $84$ cm by $60$ cm. Square tiles of equal size are laid on the floor so that no tile needs to be cut. Find (a) the largest possible side length of a tile, and (b) the number of tiles used.
Step-by-step solution
1. Step 1
  For the tiles to fit without cutting, the side length must divide both $84$ and $60$ exactly. The largest such length is $\text{HCF}(84, 60)$ .
2. Step 2
  Prime factorise: $84 = 2^{2} \times 3 \times 7$ and $60 = 2^{2} \times 3 \times 5$ .
3. Step 3
  Take common primes raised to the lower power: $\text{HCF} = 2^{2} \times 3 = 12$ .
  $\text{HCF}(84, 60) = 12$
4. Step 4
  Number of tiles $= \dfrac{\text{floor area}}{\text{tile area}} = \dfrac{84 \times 60}{12 \times 12} = \dfrac{5040}{144} = 35$ .
Answer
(a) $12$ cm (b) $35$ tiles.
Examiner tip
The 2023 examiner report flags that many candidates jumped to LCM here, misled by the word 'largest'. The tile must fit inside both dimensions — that is HCF, not LCM.
10LCM word problem — when do periodic events coincide?
ExtendedWord problem• Adapted from 0580/42 May/Jun 2023 Q3• LCM, word problem, context
Question
Three warning lights flash at intervals of $4$ , $6$ and $9$ seconds. They flash together at $9{:}00{:}00$ am. (a) When will they next flash together? (b) How many times do all three flash together in the first $5$ minutes, excluding the start?
Step-by-step solution
1. Step 1
  All three flash together every $\text{LCM}(4, 6, 9)$ seconds.
2. Step 2
  Prime factorise: $4 = 2^{2}$ , $6 = 2 \times 3$ , $9 = 3^{2}$ .
3. Step 3
  LCM takes every prime raised to its highest power: $\text{LCM} = 2^{2} \times 3^{2} = 36$ seconds.
  $\text{LCM}(4, 6, 9) = 36$
4. Step 4
  (a) The next coincidence is $36$ seconds after $9{:}00{:}00$ , i.e. $9{:}00{:}36$ am.
5. Step 5
  (b) Five minutes $= 300$ seconds. The number of coincidences in that window is $\lfloor 300 \div 36 \rfloor = 8$ (since $36 \times 8 = 288 \le 300 < 324 = 36 \times 9$ ).
Answer
(a) $9{:}00{:}36$ am (b) $8$ times.
Examiner tip
Examiner reports show candidates frequently list multiples by hand and miscount or stop early. Prime-factorise and use the LCM rule — quicker and far more reliable under time pressure.
11Use the HCF × LCM identity to find a missing number
ChallengeMulti-step problem• Adapted from 0580/42 Oct/Nov 2023 Q8• HCF, LCM, identity, stretch
Question
Two positive integers have $\text{HCF} = 12$ and $\text{LCM} = 360$ . One of the numbers is $60$ . Find the other number.
Step-by-step solution
1. Step 1
  Use the identity $a \times b = \text{HCF}(a,b) \times \text{LCM}(a,b)$ , which holds for any two positive integers.
  $a \times b = \text{HCF} \times \text{LCM}$
2. Step 2
  Substitute the known values: $60 \times b = 12 \times 360$ .
  $60 \times b = 4320$
3. Step 3
  Solve: $b = \dfrac{4320}{60} = 72$ .
4. Step 4
  Verify by prime factorisation. $60 = 2^{2} \times 3 \times 5$ and $72 = 2^{3} \times 3^{2}$ . $\text{HCF} = 2^{2} \times 3 = 12$ ✓ and $\text{LCM} = 2^{3} \times 3^{2} \times 5 = 360$ ✓.
Answer
The other number is $72$ .
Examiner tip
Candidates who do not know the HCF × LCM identity often try trial-and-error and time out. Memorise the identity — it converts a 4-mark stretch question into a single-step calculation.
12Show that the sum of three consecutive integers is divisible by 3
ChallengeShow that / prove• Adapted from 0580/42 May/Jun 2024 Q23• proof, show that, divisibility, algebra
Question
Let $n$ be any integer. Show algebraically that the sum $n + (n+1) + (n+2)$ is always a multiple of $3$ .
Step-by-step solution
1. Step 1
  Write the three consecutive integers in terms of $n$ : $n$ , $n+1$ , $n+2$ .
2. Step 2
  Add them together: $n + (n+1) + (n+2) = 3n + 3$ .
  $n + (n+1) + (n+2) = 3n + 3$
3. Step 3
  Factor out $3$ : $3n + 3 = 3(n + 1)$ .
  $3n + 3 = 3(n + 1)$
4. Step 4
  Since $n$ is an integer, $(n+1)$ is also an integer. Therefore $3(n+1)$ is $3$ multiplied by an integer, i.e. a multiple of $3$ . QED.
Answer
$n + (n+1) + (n+2) = 3(n+1)$ , which is $3$ times an integer, hence divisible by $3$ .
Examiner tip
The 2024 examiner report emphasises that 'show that' answers must end with an explicit conclusion line — e.g. ' $3(n+1)$ is a multiple of $3$ because $(n+1)$ is an integer.' Without that final statement, the proof is incomplete and loses the conclusion mark.

Key Formulae — Number Theory

The formulae you need to memorise for number theory on the Cambridge IGCSE 0580 paper, with every variable defined in plain English and a note on when to use it.

HCF · LCM identity for two integers
$a \times b = \text{HCF}(a,b) \times \text{LCM}(a,b)$
$a, b$
any two positive integers
$\text{HCF}(a,b)$
highest common factor (greatest common divisor) of $a$ and $b$
$\text{LCM}(a,b)$
lowest common multiple of $a$ and $b$
When to use
Use this identity to find one of HCF or LCM when the other is known. It saves time on questions that give you the product $a \times b$ and either the HCF or LCM.
Example
If $a=12$ , $b=18$ and HCF $=6$ , then LCM $= \dfrac{12 \times 18}{6} = 36$ .
Divisibility tests for 3 and 9
$n\ \text{is divisible by}\ 3 \iff \text{digit sum of}\ n\ \text{is divisible by}\ 3$
$n$
any positive integer
When to use
Apply this whenever you need to test factors quickly without long division. The same rule with $9$ in place of $3$ tests for divisibility by $9$ .
Example
$8{,}253$ : digit sum $= 8+2+5+3 = 18$ , divisible by both $3$ and $9$ .
Divisibility test for 4
$n\ \text{is divisible by}\ 4 \iff \text{the last two digits of}\ n\ \text{form a multiple of}\ 4$
$n$
any positive integer
When to use
Quickly testing whether a large number divides by $4$ — useful when the prime factorisation method is overkill.
Example
$3{,}716$ : last two digits $16 = 4 \times 4$ , so divisible by $4$ .
Number of factors of a number
$\text{If}\ n = p_{1}^{a_{1}} p_{2}^{a_{2}} \dots p_{k}^{a_{k}},\ \text{then}\ \tau(n) = (a_{1}+1)(a_{2}+1)\dots(a_{k}+1)$
$p_{i}$
distinct prime factors of $n$
$a_{i}$
the exponent of $p_{i}$ in the prime factorisation
$\tau(n)$
the total number of positive factors of $n$
When to use
Use this to count factors quickly without listing them — appears in extension questions on number patterns.
Example
$360 = 2^{3} \times 3^{2} \times 5^{1}$ has $(3+1)(2+1)(1+1) = 24$ factors.
Reciprocal of a non-zero number
$\text{reciprocal of}\ x = \dfrac{1}{x},\ x \neq 0$
$x$
any non-zero real number
When to use
Reciprocals appear in division questions, in defining the multiplicative inverse, and in proportion problems.
Example
Reciprocal of $\dfrac{3}{4}$ is $\dfrac{4}{3}$ .

Key Definitions and Keywords — Number Theory

Definitions to memorise and the exact keywords mark schemes credit for number theory answers — sharpened from recent examiner reports for the 2026 0580 sitting.

Natural number
Examiner keyword
A positive integer used for counting: $1, 2, 3, 4, \dots$ . Cambridge IGCSE 0580 follows the convention that $0$ is not a natural number.
Example
$7$ and $25$ are natural numbers; $-3$ , $0$ and $\frac{1}{2}$ are not.
Integer
Examiner keyword
Any whole number, positive, negative, or zero: $\dots, -2, -1, 0, 1, 2, \dots$ .
Example
$-7$ , $0$ and $42$ are integers; $\frac{2}{3}$ is not.
Prime number
Examiner keyword
An integer greater than $1$ whose only positive factors are $1$ and itself. Note that $1$ is not prime.
Example
The first six primes are $2, 3, 5, 7, 11, 13$ .
Composite number
A positive integer greater than $1$ that has at least one factor other than $1$ and itself.
Example
$12$ is composite ( $12 = 2 \times 2 \times 3$ ).
Square number
Examiner keyword
An integer of the form $n^{2}$ where $n$ is a positive integer.
Example
$1, 4, 9, 16, 25, 36, \dots$
Cube number
Examiner keyword
An integer of the form $n^{3}$ where $n$ is a positive integer.
Example
$1, 8, 27, 64, 125, \dots$
Triangular number
A number of the form $\frac{n(n+1)}{2}$ for $n = 1, 2, 3, \dots$ — the sum of the first $n$ positive integers.
Example
$1, 3, 6, 10, 15, 21, 28, \dots$
Rational number
Examiner keyword
A number that can be written as $\dfrac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$ . Every terminating or recurring decimal is rational.
Example
$\frac{3}{7}$ , $0.25$ , $1.\overline{3}$ and $\sqrt{16}=4$ .
Irrational number
Examiner keyword
A real number that cannot be written as a fraction of two integers. Its decimal expansion neither terminates nor recurs.
Example
$\sqrt{2}$ , $\sqrt{17}$ , $\pi$ , $e$ .
Factor (divisor)
An integer that divides another integer with no remainder.
Example
The factors of $12$ are $1, 2, 3, 4, 6, 12$ .
Related: multiple, hcf
Multiple
The result of multiplying an integer by another integer.
Example
Multiples of $4$ are $4, 8, 12, 16, 20, \dots$
Related: factor, lcm
Highest common factor (HCF)
Examiner keyword
The largest positive integer that divides each of two or more given integers.
Example
$\text{HCF}(18, 24) = 6$ .
Related: lcm, factor
Lowest common multiple (LCM)
Examiner keyword
The smallest positive integer that is a multiple of each of two or more given integers.
Example
$\text{LCM}(4, 6) = 12$ .
Related: hcf, multiple
Reciprocal
Examiner keyword
The reciprocal of a non-zero number $x$ is $\dfrac{1}{x}$ . The product of any number and its reciprocal is $1$ .
Example
Reciprocal of $5$ is $\dfrac{1}{5}$ . Reciprocal of $\dfrac{2}{3}$ is $\dfrac{3}{2}$ .
Prime factorisation
Examiner keyword
Expressing a positive integer as a product of prime numbers. By the Fundamental Theorem of Arithmetic this product is unique up to ordering.
Example
$360 = 2^{3} \times 3^{2} \times 5$ .

Common Mistakes and Misconceptions — Number Theory

The traps other students keep falling into on number theory questions — taken from recent Cambridge IGCSE 0580 examiner reports and mark schemes — and how to avoid them.

✕Including $1$ in a list of prime numbers
0580/22 Jun 2022 — examiner report
Why it happens
Students remember the rule "primes have factors $1$ and themselves" but forget that this requires exactly two distinct factors — and $1$ has only one.
How to avoid it
Memorise the formal definition: a prime is an integer greater than $1$ with exactly two positive factors. Cross-check by listing factors.
✕Marking a negative number as prime
0580/12 Oct/Nov 2023 — examiner report
Why it happens
When a question asks for primes from a mixed list (with negatives), students miss the requirement that primes are positive.
How to avoid it
Read the definition every time: primes are positive integers greater than 1. If it's negative, immediately exclude it.
✕Swapping the HCF and LCM rules in prime-factorisation questions
0580/42 May/Jun 2024 — examiner report Q2
Why it happens
The rules sound similar — "common primes" for both — and under exam pressure students apply the wrong power.
How to avoid it
Anchor it: HCF = lower power, LCM = larger power. The HCF must be small (it has to fit inside both numbers), so the powers must be small.
✕Leaving prime factorisation as a long product instead of in index form
0580/12 Oct/Nov 2022 — examiner report Q4
Why it happens
Students stop as soon as they get to primes ( $2 \times 2 \times 2 \times 3 \times 3 \times 5$ ) and forget the question asked for index form.
How to avoid it
Always read the final part of the question. If it says "in index form", group the repeats: $2^{3} \times 3^{2} \times 5$ .
✕Writing an irrational answer as a rounded decimal
0580/42 Feb/Mar 2023 — examiner report
Why it happens
Students reach for the calculator and write something like $\sqrt{17} \approx 4.123$ , treating it as a final answer.
How to avoid it
If the question requires the exact irrational form, leave it as a surd. A rounded decimal turns the answer rational and loses the accuracy mark.
✕Calling a recurring decimal irrational
Why it happens
Students think "never-ending" implies irrational, but they confuse recurring (repeats) with irrational (non-repeating non-terminating).
How to avoid it
Rule: every recurring decimal can be written as a fraction, so every recurring decimal is rational. $0.\overline{3} = \dfrac{1}{3}$ .
✕Flipping a mixed number directly without converting first
Why it happens
The reciprocal of $2\frac{1}{4}$ is wrongly written as $\frac{1}{2}\frac{4}{1}$ or $\frac{1}{2\frac{1}{4}}$ left unsimplified.
How to avoid it
Always convert mixed to improper first: $2\frac{1}{4} = \frac{9}{4}$ , then flip to get $\frac{4}{9}$ .
✕Calling $0$ a natural number
Cambridge IGCSE 0580 syllabus 2025-2027 — Section 1.1
Why it happens
Different textbooks use different conventions; some include $0$ in the naturals.
How to avoid it
Cambridge IGCSE 0580 follows the convention that natural numbers start at $1$ . $0$ is an integer but not natural.
✕Writing $\sqrt{16}$ as irrational
0580/22 May/Jun 2023 — examiner report
Why it happens
Surds look irrational, so students assume any expression with $\sqrt{\ }$ is irrational without simplifying.
How to avoid it
Always simplify first. $\sqrt{16} = 4$ is an integer, hence rational. Only $\sqrt{n}$ where $n$ is not a perfect square is irrational.

Number Theory — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Number Theory

Short Notes - Number Theory

Detailed Study Notes

At a glance

What you’ll learn

The four families of numbers

Prime numbers — recognition

Square and cube numbers

Reciprocals

Ordering and the number line

Quick recap

Memorise this

How it’s examined

Take this whole topic with you

Question patterns to master — Number Theory

1Classify a list of numbers (rational, irrational, integer, prime)

2Write a number as a product of prime factors

3Find the HCF and LCM using prime factorisation

4Determine whether an expression is rational or irrational

5Use a divisibility test to identify factors

6Identify square, cube and triangular numbers from a sequence

7Find the reciprocal of a mixed expression

8List all factors of an integer and identify the primes

9HCF word problem — the largest tile that fits without cutting

10LCM word problem — when do periodic events coincide?

11Use the HCF × LCM identity to find a missing number

12Show that the sum of three consecutive integers is divisible by 3

HCF · LCM identity for two integers

Divisibility tests for 3 and 9

Divisibility test for 4

Number of factors of a number

Reciprocal of a non-zero number

Natural number

Integer

Prime number

Composite number

Square number

Cube number

Triangular number

Rational number

Irrational number

Factor (divisor)

Multiple

Highest common factor (HCF)

Lowest common multiple (LCM)

Reciprocal

Prime factorisation

✕Including 111 in a list of prime numbers

✕Marking a negative number as prime

✕Swapping the HCF and LCM rules in prime-factorisation questions

✕Leaving prime factorisation as a long product instead of in index form

✕Writing an irrational answer as a rounded decimal

✕Calling a recurring decimal irrational

✕Flipping a mixed number directly without converting first

✕Calling 000 a natural number

✕Writing 16\sqrt{16}16​ as irrational

Number Theory — frequently asked questions

✕Including $1$ in a list of prime numbers

✕Calling $0$ a natural number

✕Writing $\sqrt{16}$ as irrational