What are integers in the context of the Edexcel IGCSE Mathematics curriculum?

In the Edexcel IGCSE Mathematics curriculum, integers are defined as whole numbers that can be positive, negative, or zero. They do not include fractions or decimals. Integers are a fundamental part of the Numbers and the Number System topic, and understanding them is crucial for solving various mathematical problems.

How do you add and subtract integers?

To add integers, if they have the same sign, you add their absolute values and keep the sign. If they have different signs, subtract the smaller absolute value from the larger one and take the sign of the number with the larger absolute value. For subtraction, you add the opposite of the integer being subtracted. This method is consistent with the rules taught in the Edexcel IGCSE Mathematics curriculum.

What is the difference between integers and whole numbers?

Whole numbers are a subset of integers. Whole numbers include all positive integers and zero, but do not include negative numbers. In contrast, integers include all whole numbers as well as their negative counterparts. This distinction is important in the Numbers and the Number System topic of the Edexcel IGCSE Mathematics syllabus.

How are integers used in real-life applications?

Integers are used in various real-life applications such as calculating temperatures (above and below zero), financial transactions (credits and debits), and elevation levels (above and below sea level). Understanding integers is essential for interpreting data and solving practical problems, which is emphasized in the Edexcel IGCSE Mathematics curriculum.

What are the properties of integers that are important for solving equations?

Key properties of integers include closure (the sum or product of any two integers is an integer), commutativity (order does not matter in addition or multiplication), associativity (grouping does not affect the sum or product), and distributivity (multiplication distributes over addition). These properties are crucial for solving equations and are covered in the Edexcel IGCSE Mathematics syllabus under the Numbers and the Number System topic.

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

Why it happens: Students remember 'primes have factors 1 and themselves' but forget the requirement that there be EXACTLY TWO factors — and 1 has only one. How to avoid it: Quote the formal definition: a prime is an integer GREATER than 1 with exactly two positive factors. Cross-check by listing factors. Source: 4MA1/1H Jan 2023 — examiner report Q2

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

Why it happens: Students stop as soon as the primes appear and miss the question's instruction to use index form. How to avoid it: Always reread the question. If it says 'in index form', group repeats: write 2 × 2 × 2 × 3 × 3 as 2^3 × 3^2. Source: 4MA1/1H May/Jun 2023 — examiner report Q3

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

Why it happens: The rules sound similar — 'common primes' for both — and under exam pressure students use the wrong power. How to avoid it: Anchor it: HCF = LOWER power of common primes; LCM = HIGHER power of all primes appearing in either. The HCF must be small (it has to fit inside both numbers). Source: 4MA1/2H May/Jun 2024 — examiner report Q5

Why is "Choosing HCF when an LCM problem is asked (or vice versa)" a common mistake in Integers?

Why it happens: Real-life problems hide which one is needed. Students just pick the easier calculation. How to avoid it: Test the question type: 'When do they meet again?' or 'how often do both happen?' = LCM. 'Largest equal share?' or 'biggest box that fits both?' = HCF. Source: 4MA1/2H May/Jun 2024 — examiner report Q11

Why is "Forgetting that subtracting a negative gives a positive" a common mistake in Integers?

Why it happens: Many candidates write 9 - (-20) = -11 instead of 9 + 20 = 29. How to avoid it: Replace 'subtract negative' with 'add positive' BEFORE evaluating: 9 - (-20) = 9 + 20 = 29. Source: 4MA1/1H — examiner reports 2022-2024

Why is "Listing $0$ as a factor of a number" a common mistake in Integers?

Why it happens: Students confuse 'multiple' with 'factor' (every integer is a factor of zero, but zero is never a factor of any non-zero integer). How to avoid it: Factors of n are positive integers that divide n. Always start your list at 1, never at 0.

Why is "Writing $(-3)^{2} = -9$" a common mistake in Integers?

Why it happens: Students apply the negative sign AFTER squaring, treating (-3)^2 as -3^2. How to avoid it: (-3)^2 means (-3) × (-3) = 9. The brackets matter: -3^2 = -9 (only the 3 is squared) but (-3)^2 = 9. Source: 4MA1/1H Jan 2024 — examiner report

Why is "Forgetting that $1$ is a square AND a cube AND a sixth power" a common mistake in Integers?

Why it happens: Students think 'square numbers start at 4' or 'cube numbers start at 8' from memory. How to avoid it: Always include 1 unless the question explicitly says 'greater than 1'. 1 = 1^2 = 1^3 = 1^n for any n. Source: 4MA1/1H Oct/Nov 2023 — examiner report Q2

Numbers and the Number SystemPearson Edexcel IGCSE Mathematics 4MA1 Higher

Integers

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Integers Study Notes — Edexcel IGCSE 4MA1 Higher Tier (2026 onwards)

Foundation for the whole Number topic. Master types of integers, factors, multiples, primes, HCF and LCM — plus the divisibility rules and word-problem cues that examiners reward.

What you’ll learn

Mapped to the Pearson Edexcel IGCSE 4MA1 syllabus (2026 onwards).

1.1 — Identify and use natural numbers, integers, even and odd numbers, prime numbers, square and cube numbers.
1.1 — Use the terms factor, multiple, common factor, common multiple.
1.1 — Find the HCF and LCM of two or more numbers.
1.1 — Write integers as a product of prime factors using index notation.
1.1 — Use BIDMAS/BODMAS for order of operations including negative numbers.

Types of integers — what each one is

Natural, integer, prime, composite, square, cube — Edexcel exam vocabulary.

Edexcel 4MA1 expects you to use precise vocabulary. Mark schemes credit specific keywords.

Natural numbers ( $\mathbb{N}$ ). $1, 2, 3, 4, \ldots$ — positive integers used for counting. $0$ is NOT a natural number under the Edexcel convention.

Integers ( $\mathbb{Z}$ ). $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$ — every whole number, positive, negative, or zero.

Even and odd integers. Even = divisible by $2$ ; odd = not divisible by $2$ . Note $0$ is even.

Prime numbers. Integers $> 1$ with EXACTLY two positive factors ( $1$ and itself). The first ten primes: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29$ . $2$ is the only even prime. $1$ is NOT prime.

Composite numbers. Positive integers $> 1$ that have more than two factors. Equivalently: positive integers $> 1$ that aren't prime.

Square numbers. $n^{2}$ : $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, \ldots$ . Memorise to $15^{2} = 225$ .

Cube numbers. $n^{3}$ : $1, 8, 27, 64, 125, 216, 343, 512, 729, 1000$ . Memorise to $10^{3}$ .

Worked qualitative. Why is $1$ not a prime number?

A prime needs EXACTLY two distinct factors.
$1$ has only ONE factor (itself).
Excluding $1$ from primes also makes the Fundamental Theorem of Arithmetic clean (every integer has a unique prime factorisation).

Edexcel tip. State 'integer' explicitly when asked. Many candidates write 'whole number' instead — Edexcel mark schemes accept both, but 'integer' is the technical term and is always rewarded.

Natural: $1, 2, 3, \ldots$ (no $0$ ).
Integer: any whole number including $0$ and negatives.
Prime: exactly 2 factors. $1$ excluded.
Square + cube — memorise common ones.
$0$ is even.

Prime factorisation — the foundation for HCF and LCM

Every integer $> 1$ has a unique product of primes. Use a factor tree. Always give in INDEX form.

Why prime factorisation? Two numbers' prime factorisations make HCF and LCM trivial to compute. Almost every Number 1 question expects you to factorise first.

Method 1 — Factor tree.

Start with the number. Divide by the smallest prime that fits. Continue branching until all leaves are prime.

Example: $360$

Read off all the primes: $360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5$ .

In INDEX form: $360 = 2^{3} \times 3^{2} \times 5$ .

Green leaves are primes; collect them into index form.

Method 2 — Repeated division.

Same answer.

Worked qualitative. What's special about index form vs the long product?

Long product: $2 \times 2 \times 2 \times 3 \times 3 \times 5$ .
Index form: $2^{3} \times 3^{2} \times 5^{1}$ .
Edexcel mark schemes have separate marks for METHOD (factor tree shown) and ANSWER in INDEX FORM.
Stopping at the long product = lose the A1.

Edexcel tip. When the question says "in index form" and you don't, you lose the accuracy mark. Always group repeats AND drop the $^1$ on lone primes (write $5$ , not $5^{1}$ ).

Factor tree or repeated division.
Always give answer in INDEX form.
Group repeats: $2 \times 2 \times 2 = 2^{3}$ .
Lone primes: drop the $^1$ .
M1 for method, A1 for index form.

HCF and LCM — the rules and identity

HCF: lower powers of common primes. LCM: higher powers of all primes. Memorise the product identity.

Once you've prime-factorised both numbers, computing HCF and LCM is mechanical.

HCF (Highest Common Factor) rule. Take each prime that appears in BOTH factorisations, raised to the LOWER power.

LCM (Lowest Common Multiple) rule. Take EVERY prime that appears in EITHER factorisation, raised to the HIGHER power.

Worked example. Find HCF and LCM of $A = 2^{4} \times 3^{2} \times 5$ and $B = 2^{2} \times 3^{3} \times 7$ .

Prime	$A$	$B$	HCF (lower)	LCM (higher)
$2$	$2^{4}$	$2^{2}$	$2^{2}$	$2^{4}$
$3$	$3^{2}$	$3^{3}$	$3^{2}$	$3^{3}$
$5$	$5^{1}$	—	—	$5^{1}$
$7$	—	$7^{1}$	—	$7^{1}$

$\text{HCF} = 2^{2} \times 3^{2} = 4 \times 9 = 36$ .
$\text{LCM} = 2^{4} \times 3^{3} \times 5 \times 7 = 16 \times 27 \times 5 \times 7 = 15{,}120$ .

HCF uses only the shared overlap; LCM uses every prime in the diagram.

The product identity (memorise — NOT on Edexcel formula sheet):

$a \times b = \text{HCF}(a, b) \times \text{LCM}(a, b)$

Useful when given 3 of the 4 quantities.

Worked qualitative. When do you use HCF vs LCM in word problems?

Phrase	Use
"When will they next coincide?"	LCM (find when both repeat together)
"Greatest length / largest box / biggest equal share"	HCF (find biggest divisor of both)
"How many groups of equal size?"	HCF
"How long until they all flash together?"	LCM

Edexcel tip. Show the prime factorisations FIRST, then circle/highlight common primes. This earns method marks even if final arithmetic is wrong.

HCF: common primes, LOWER power.
LCM: all primes, HIGHER power.
Identity: $ab = \text{HCF} \times \text{LCM}$ .
When-they-meet-again → LCM.
Biggest equal share → HCF.

Divisibility tests — quick checks

Memorise these rules. They save time on Paper 1H non-calculator questions.

Edexcel Paper 1H has no calculator. These rules let you test factors instantly.

Divisor	Test
$2$	Last digit even ( $0, 2, 4, 6, 8$ )
$3$	Digit sum divisible by $3$
$4$	Last two digits form a multiple of $4$
$5$	Last digit is $0$ or $5$
$6$	Divisible by both $2$ AND $3$
$8$	Last three digits form a multiple of $8$
$9$	Digit sum divisible by $9$
$10$	Last digit is $0$
$11$	Alternating digit sum is divisible by $11$

Example: is $7{,}524$ divisible by $9$ ?

Digit sum: $7 + 5 + 2 + 4 = 18$ .
$18 \div 9 = 2$ — yes.

Example: is $7{,}524$ divisible by $11$ ?

Alternating sum: $7 - 5 + 2 - 4 = 0$ .
$0$ is a multiple of $11$ — yes.

Worked qualitative. A 5-digit number $4\,a\,2\,7\,5$ is divisible by $9$ . Find $a$ .

Digit sum: $4 + a + 2 + 7 + 5 = 18 + a$ .
For divisibility by $9$ : $18 + a$ must be a multiple of $9$ .
Since $0 \le a \le 9$ : $a = 0$ gives $18$ , $a = 9$ gives $27$ .
Both work — answer: $a = 0$ or $a = 9$ .

Edexcel tip. State the rule before applying it. Mark schemes for digit-puzzles award M1 for the rule and A1 for the answer.

$3, 9$ — digit sum.
$4$ — last two digits.
$8$ — last three digits.
$11$ — alternating sum.
State the rule for the M1 mark.

Working with negative integers

Even powers stay positive. Subtracting a negative is adding. BIDMAS rules unchanged.

Negative integers appear throughout 4MA1 — coordinates, temperatures, finance.

The four arithmetic rules.

Operation	Result
$+ \times +$	$+$
$- \times -$	$+$
$+ \times -$	$-$
$- \times +$	$-$

Same rules apply to division.

Powers of negatives.

$(-2)^{2} = +4$ (even exponent → positive).
$(-2)^{3} = -8$ (odd exponent → negative).
General: $(-x)^{n} = +x^n$ if $n$ even, $-x^n$ if $n$ odd.

Watch the brackets.

$(-3)^{2} = 9$ .
$-3^{2} = -9$ (just the $3$ is squared).

Subtracting a negative.

Replace 'subtract negative' with 'add positive':

$9 - (-20) = 9 + 20 = 29$ .

BIDMAS still applies:

Brackets
Indices (powers/roots)
Division/Multiplication (left to right)
Addition/Subtraction (left to right)

Worked qualitative. Calculate $(-2)^{4} + (-3) \times (-5) - (-7)$ .

Powers first: $(-2)^{4} = 16$ .
Multiplication next: $(-3) \times (-5) = 15$ .
Substitute: $16 + 15 - (-7) = 16 + 15 + 7 = 38$ .

Edexcel tip. Underline negative numbers in the question to remind yourself they're there. The double-negative trap is the single most common arithmetic error in Paper 1H.

$- \times - = +$ .
Even power of $-x$ is positive.
$(-3)^2 = 9$ , but $-3^2 = -9$ .
Subtract negative = add positive.
BIDMAS applies as usual.

How it’s examined

Integers appear on EVERY 4MA1 paper (Paper 1H + Paper 2H). Typical formats: prime factorise (2-3 marks), HCF/LCM (3-4 marks), digit puzzle (2-3 marks), word problem with HCF or LCM (3-5 marks). Examiner reports flag (1) candidates not giving prime factorisation in index form, (2) using HCF instead of LCM in 'when do they meet?' problems, and (3) the double-negative arithmetic trap on Paper 1H.

Sources: Pearson Edexcel International GCSE Mathematics A 4MA1 specification (2026 onwards); 4MA1 Examiner Reports — Jan and May/Jun 2022-2024 series; Past Papers — 4MA1/1H May/Jun 2023, 4MA1/2H May/Jun 2024, 4MA1/1H Oct/Nov 2023. Last reviewed 2026-05-07.

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Worked examples, formulae, definitions and the mistakes examiners flag — everything you need to push from a pass to an A*.

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Step-by-step worked examples — Integers

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

1Write a number as a product of prime factors in index form
Foundation• Adapted from 4MA1/1H May/Jun 2023 Q3• prime factorisation, index form
Question
Write $504$ as a product of its prime factors. Give your answer in index form.
Step-by-step solution
1. Step 1
  Use a factor tree, dividing by the smallest prime each time. Start with $2$ : $504 \div 2 = 252$ .
2. Step 2
  Continue: $252 \div 2 = 126$ , $126 \div 2 = 63$ .
3. Step 3
  $63$ is odd, move to the next prime: $63 \div 3 = 21$ , $21 \div 3 = 7$ .
4. Step 4
  $7$ is itself prime, so stop. Collect the primes:
  $504 = 2 \times 2 \times 2 \times 3 \times 3 \times 7$
5. Step 5
  Group repeats into index form.
  $504 = 2^{3} \times 3^{2} \times 7$
Answer
$504 = 2^{3} \times 3^{2} \times 7$
Examiner tip
Edexcel mark schemes split this into M1 (correct method shown — factor tree or repeated division) and A1 (correct prime factor product in INDEX form). Stopping at $2 \times 2 \times 2 \times 3 \times 3 \times 7$ loses the A1.
2Find the HCF and LCM from prime factorisations
Higher• Adapted from 4MA1/2H May/Jun 2024 Q5• HCF, LCM, prime factorisation
Question
Two numbers $A$ and $B$ are written as products of their prime factors:\n $A = 2^{3} \times 3 \times 5^{2}$ \n $B = 2^{2} \times 3^{2} \times 7$ \nFind (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
  HCF rule: 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 $3^{1}$ ). The primes $5$ and $7$ aren't in both — exclude them.
  $\text{HCF} = 2^{2} \times 3 = 4 \times 3 = 12$
3. Step 3
  LCM rule: take EVERY prime that appears in either number, raised to the HIGHER power.
4. Step 4
  Primes used: $2^{3}$ , $3^{2}$ , $5^{2}$ , $7$ .
  $\text{LCM} = 2^{3} \times 3^{2} \times 5^{2} \times 7 = 8 \times 9 \times 25 \times 7 = 12600$
Answer
(a) $\text{HCF} = 12$ (b) $\text{LCM} = 12{,}600$
Examiner tip
The 4MA1 mark scheme awards 2 marks for each part: M1 for the correct rule applied (showing the lower/higher powers) and A1 for the answer. State the powers explicitly — examiners cannot award the M1 from a bare numerical answer.
3Use the HCF × LCM identity
Higher• HCF, LCM
Question
Two positive integers have HCF $= 8$ and LCM $= 240$ . One of the numbers is $48$ . Find the other.
Step-by-step solution
1. Step 1
  Use the identity for two integers $a$ and $b$ :
  $a \times b = \text{HCF}(a,b) \times \text{LCM}(a,b)$
2. Step 2
  Substitute: $48 \times b = 8 \times 240$ .
  $48b = 1920$
3. Step 3
  Solve for $b$ :
  $b = \frac{1920}{48} = 40$
Answer
The other number is $40$ .
Examiner tip
This identity is examined regularly on Paper 2H. Quote it explicitly to score the method mark — the formula is NOT given on the Edexcel formula sheet.
4Classify numbers as prime, square or cube
Foundation• Adapted from 4MA1/1H Oct/Nov 2023 Q2• prime, square, cube
Question
From the list $\{ 1,\ 9,\ 17,\ 21,\ 27,\ 49,\ 64 \}$ write down (a) the prime numbers, (b) the cube numbers, (c) the numbers that are both square AND cube.
Step-by-step solution
1. Step 1
  Primes: integers $> 1$ with exactly two factors. Check each: $9 = 3 \times 3$ (not prime), $17$ (only $1, 17$ — prime), $21 = 3 \times 7$ (not prime), $27 = 3^{3}$ (not prime), $49 = 7^{2}$ (not prime), $64 = 2^{6}$ (not prime). Note $1$ is NOT prime.
2. Step 2
  Cubes: $n^{3}$ for some positive integer $n$ . From the list: $1 = 1^{3}$ , $27 = 3^{3}$ , $64 = 4^{3}$ .
3. Step 3
  Both square AND cube: must be of the form $n^{6}$ . From the list, $1 = 1^{6}$ and $64 = 2^{6}$ .
Answer
(a) $17$ (b) $1, 27, 64$ (c) $1$ and $64$
Examiner tip
$1$ is a square AND a cube AND a sixth power. Edexcel reports note candidates routinely forget to include $1$ and lose the mark.
5Use divisibility rules to find a missing digit
Higher• Adapted from 4MA1/1H Jan 2024 Q4• divisibility, digit puzzles
Question
The five-digit number $7\,2\,a\,1\,8$ is divisible by $9$ . Find the value of the digit $a$ .
Step-by-step solution
1. Step 1
  Divisibility rule for $9$ : the digit sum must be a multiple of $9$ .
2. Step 2
  Digit sum: $7 + 2 + a + 1 + 8 = 18 + a$ .
3. Step 3
  $18 + a$ must be a multiple of $9$ . Since $a$ is a single digit ( $0 \le a \le 9$ ), the only options are $a = 0$ (giving $18$ ) or $a = 9$ (giving $27$ ).
4. Step 4
  If the question expects a unique answer, $a = 0$ is the smallest. (Both are mathematically valid — list both unless the question constrains.)
Answer
$a = 0$ or $a = 9$
Examiner tip
Always state the divisibility rule before computing. Mark schemes for these problems award M1 for the correct rule (digit sum) and A1 for both/single value of $a$ .
6Order of operations with negative integers
Foundation• negative numbers, BIDMAS
Question
Calculate $(-3)^{2} - 4 \times (-5) + (-2)^{3}$ .
Step-by-step solution
1. Step 1
  Apply BIDMAS. Powers first: $(-3)^{2} = 9$ (positive — even power), and $(-2)^{3} = -8$ (negative — odd power).
2. Step 2
  Multiplication next: $4 \times (-5) = -20$ .
3. Step 3
  The expression becomes $9 - (-20) + (-8) = 9 + 20 - 8$ .
4. Step 4
  Simplify: $9 + 20 - 8 = 21$ .
Answer
$21$
Examiner tip
Even powers of negatives give positives; odd powers stay negative. The double-negative trap ( $-(-20) = +20$ ) is one of the most common Paper 1H errors flagged in Edexcel reports.
7Real-life LCM word problem
Higher• Adapted from 4MA1/2H May/Jun 2024 Q11• LCM, word problem
Question
Two lighthouses flash at regular intervals. Lighthouse A flashes every $24$ seconds. Lighthouse B flashes every $36$ seconds. They flash together at $9{:}00$ am. At what time will they next flash together?
Step-by-step solution
1. Step 1
  When two events repeat at fixed intervals, they re-align at the LCM of the two intervals.
2. Step 2
  Prime factorise: $24 = 2^{3} \times 3$ , $36 = 2^{2} \times 3^{2}$ .
3. Step 3
  $\text{LCM} = 2^{3} \times 3^{2} = 8 \times 9 = 72$ seconds.
4. Step 4
  $72$ seconds $= 1$ minute $12$ seconds. So they next flash together at $9{:}01{:}12$ am.
Answer
$9{:}01{:}12$ am ( $72$ seconds after $9{:}00$ ).
Examiner tip
Recognising LCM (not HCF) is the key insight. Edexcel reports note that ~40% of candidates use HCF in this type of question — losing all the marks. RULE: 'when do they meet?' = LCM; 'how can we share equally?' = HCF.

Key Formulae — Integers

The formulae you need to memorise for integers on the Pearson Edexcel IGCSE 4MA1 paper, with every variable defined in plain English and a note on when to use it.

HCF · LCM identity
$a \times b = \text{HCF}(a,b) \times \text{LCM}(a,b)$
$a, b$
any two positive integers
$\text{HCF}(a,b)$
highest common factor of $a$ and $b$
$\text{LCM}(a,b)$
lowest common multiple of $a$ and $b$
When to use
Use this when given two of the four quantities ( $a$ , $b$ , HCF, LCM) and asked for the fourth. NOT on the Edexcel 4MA1 formula sheet — you must memorise it.
Example
If $a = 12$ , $b = 18$ and HCF $= 6$ , then LCM $= \dfrac{12 \times 18}{6} = 36$ .
Divisibility tests for 3 and 9
$n\ \text{divisible by}\ 3 \iff \text{digit sum of}\ n\ \text{divisible by}\ 3$
$n$
any positive integer
When to use
Use to test divisibility 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
$n\ \text{divisible by}\ 4 \iff \text{the last two digits of}\ n\ \text{form a multiple of}\ 4$
$n$
any positive integer
When to use
Quickly testing whether a large number divides by $4$ . Divisibility by $8$ uses last three digits.
Example
$3{,}716$ : last two digits $16 = 4 \times 4$ , so divisible by $4$ .
Total number of factors
$\text{If}\ n = p_{1}^{a_{1}} p_{2}^{a_{2}} \dots p_{k}^{a_{k}},\ \text{then}\ \tau(n) = (a_{1}+1)(a_{2}+1)\dots(a_{k}+1)$
$p_{i}$
distinct prime factors of $n$
$a_{i}$
the exponent of $p_{i}$
$\tau(n)$
the total number of positive factors of $n$
When to use
Use to count factors quickly without listing them. Useful for extension number-pattern questions on Paper 2H.
Example
$360 = 2^{3} \times 3^{2} \times 5^{1}$ has $(3+1)(2+1)(1+1) = 24$ factors.

Key Definitions and Keywords — Integers

Definitions to memorise and the exact keywords mark schemes credit for integers answers — sharpened from recent examiner reports for the 2026 Pearson Edexcel IGCSE 4MA1 sitting.

Integer
Examiner keyword
Any whole number — positive, negative, or zero. The set $\mathbb{Z} = \{ \dots, -2, -1, 0, 1, 2, \dots \}$ .
Example
$-7$ , $0$ and $42$ are integers; $\frac{2}{3}$ is not.
Natural number
Examiner keyword
A positive integer used for counting: $1, 2, 3, \dots$ . Edexcel 4MA1 follows the convention that $0$ is NOT a natural number.
Example
$7$ and $25$ are natural; $0$ and $-3$ are not.
Prime number
Examiner keyword
An integer greater than $1$ whose only positive factors are $1$ and itself. The number $1$ is NOT prime (it has only one factor).
Example
$2, 3, 5, 7, 11, 13, 17, \ldots$
Composite number
A positive integer greater than $1$ with at least one factor other than $1$ and itself.
Example
$12 = 2 \times 2 \times 3$ is composite.
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
$1, 8, 27, 64, 125, \dots$
Factor (divisor)
An integer that divides another integer with no remainder.
Example
Factors of $12$ : $1, 2, 3, 4, 6, 12$ .
Related: multiple, hcf
Multiple
The result of multiplying an integer by another integer.
Example
Multiples of $4$ : $4, 8, 12, 16, 20, \dots$
Related: factor, lcm
Highest common factor (HCF)
Examiner keyword
The largest positive integer that divides each of two or more given integers exactly. Sometimes called the GCD (greatest common divisor).
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
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
$360 = 2^{3} \times 3^{2} \times 5$ .
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 $\dfrac{3}{2}$ .

Common Mistakes and Misconceptions — Integers

The traps other students keep falling into on integers questions — taken from recent Pearson Edexcel IGCSE 4MA1 examiner reports and mark schemes — and how to avoid them.

✕Including $1$ in a list of prime numbers
4MA1/1H Jan 2023 — examiner report Q2
Why it happens
Students remember 'primes have factors $1$ and themselves' but forget the requirement that there be EXACTLY TWO factors — and $1$ has only one.
How to avoid it
Quote the formal definition: a prime is an integer GREATER than $1$ with exactly two positive factors. Cross-check by listing factors.
✕Leaving prime factorisation as a long product instead of in index form
4MA1/1H May/Jun 2023 — examiner report Q3
Why it happens
Students stop as soon as the primes appear and miss the question's instruction to use index form.
How to avoid it
Always reread the question. If it says 'in index form', group repeats: write $2 \times 2 \times 2 \times 3 \times 3$ as $2^{3} \times 3^{2}$ .
✕Swapping the HCF and LCM rules in prime-factorisation questions
4MA1/2H May/Jun 2024 — examiner report Q5
Why it happens
The rules sound similar — 'common primes' for both — and under exam pressure students use the wrong power.
How to avoid it
Anchor it: HCF = LOWER power of common primes; LCM = HIGHER power of all primes appearing in either. The HCF must be small (it has to fit inside both numbers).
✕Choosing HCF when an LCM problem is asked (or vice versa)
4MA1/2H May/Jun 2024 — examiner report Q11
Why it happens
Real-life problems hide which one is needed. Students just pick the easier calculation.
How to avoid it
Test the question type: 'When do they meet again?' or 'how often do both happen?' = LCM. 'Largest equal share?' or 'biggest box that fits both?' = HCF.
✕Forgetting that subtracting a negative gives a positive
4MA1/1H — examiner reports 2022-2024
Why it happens
Many candidates write $9 - (-20) = -11$ instead of $9 + 20 = 29$ .
How to avoid it
Replace 'subtract negative' with 'add positive' BEFORE evaluating: $9 - (-20) = 9 + 20 = 29$ .
✕Listing $0$ as a factor of a number
Why it happens
Students confuse 'multiple' with 'factor' (every integer is a factor of zero, but zero is never a factor of any non-zero integer).
How to avoid it
Factors of $n$ are positive integers that divide $n$ . Always start your list at $1$ , never at $0$ .
✕Writing $(-3)^{2} = -9$
4MA1/1H Jan 2024 — examiner report
Why it happens
Students apply the negative sign AFTER squaring, treating $(-3)^{2}$ as $-3^{2}$ .
How to avoid it
$(-3)^{2}$ means $(-3) \times (-3) = 9$ . The brackets matter: $-3^{2} = -9$ (only the $3$ is squared) but $(-3)^{2} = 9$ .
✕Forgetting that $1$ is a square AND a cube AND a sixth power
4MA1/1H Oct/Nov 2023 — examiner report Q2
Why it happens
Students think 'square numbers start at $4$ ' or 'cube numbers start at $8$ ' from memory.
How to avoid it
Always include $1$ unless the question explicitly says 'greater than $1$ '. $1 = 1^{2} = 1^{3} = 1^{n}$ for any $n$ .

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

Get a report showing which sub-topics you've nailed and which ones still need work.

Video lesson

Short walkthrough of the concepts students most often get stuck on.

Integers — frequently asked questions

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

Integers

Short Study Notes in the form of Slides

Short Study Notes — Integers

Detailed Notes

What you’ll learn

How it’s examined

1Write a number as a product of prime factors in index form

2Find the HCF and LCM from prime factorisations

3Use the HCF × LCM identity

4Classify numbers as prime, square or cube

5Use divisibility rules to find a missing digit

6Order of operations with negative integers

7Real-life LCM word problem

HCF · LCM identity

Divisibility tests for 3 and 9

Divisibility test for 4

Total number of factors

Integer

Natural number

Prime number

Composite number

Square number

Cube number

Factor (divisor)

Multiple

Highest common factor (HCF)

Lowest common multiple (LCM)

Prime factorisation

Reciprocal

✕Including 111 in a list of prime numbers

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

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

✕Choosing HCF when an LCM problem is asked (or vice versa)

✕Forgetting that subtracting a negative gives a positive

✕Listing 000 as a factor of a number

✕Writing (−3)2=−9(-3)^{2} = -9(−3)2=−9

✕Forgetting that 111 is a square AND a cube AND a sixth power

Practice questions

Past paper style quiz

Assess your understanding

Video lesson

Integers — frequently asked questions

✕Including $1$ in a list of prime numbers

✕Listing $0$ as a factor of a number

✕Writing $(-3)^{2} = -9$

✕Forgetting that $1$ is a square AND a cube AND a sixth power