Integers are the starting point for the entire Edexcel IGCSE Mathematics Specification A course, and they are among the most commonly examined topics on both papers
Every number skill you use elsewhere in the 4MA1 specification rests on integer arithmetic. Factors feed into algebraic factorisation. Multiples underpin work with fractions and LCM. Prime factorisation connects to HCF calculations and even to some Higher tier proofs. The edexcel igcse integers topic is the foundation that supports everything above it, and the exam tests it in contexts that range from pure number theory to applied word problems.
Key Facts
| Term | What it means | Example |
|---|---|---|
| Integer | A whole number, positive, negative or zero | -3, 0, 7, 142 |
| Place value | The value of a digit determined by its position | In 4723, the 7 is worth 700 |
| Factor | A number that divides exactly into another | Factors of 12: 1, 2, 3, 4, 6, 12 |
| Multiple | The result of multiplying an integer by a whole number | Multiples of 5: 5, 10, 15, 20, ... |
| Prime number | A number with exactly two factors: 1 and itself | 2, 3, 5, 7, 11, 13, ... |
| Odd number | An integer not divisible by 2 | 1, 3, 5, 7, ... |
| Even number | An integer divisible by 2 | 0, 2, 4, 6, ... |
Directed numbers
The specification requires you to use directed numbers in practical situations, such as temperature changes, bank balances and altitude. The integers explained approach begins with the rules for combining positive and negative values, because these rules reappear in algebra, coordinates and graph work.
Addition and subtraction:
- Adding a positive number moves right on the number line: -4 + 7 = 3
- Adding a negative number moves left: 5 + (-3) = 5 - 3 = 2
- Subtracting a negative number is the same as adding its positive: 6 - (-2) = 6 + 2 = 8
Multiplication and division:
- Positive x positive = positive: 4 x 3 = 12
- Negative x negative = positive: -4 x -3 = 12
- Positive x negative = negative: 4 x -3 = -12
- The same sign rules apply to division
Think of it as counting the number of negative signs: an even number of negatives gives a positive result; an odd number gives a negative result.
Order of operations (BIDMAS)
The specification states that candidates must use brackets and the hierarchy of operations. The order is: Brackets, Indices (powers), Division and Multiplication (left to right), Addition and Subtraction (left to right).
Worked example: Calculate 3 + 4 x (8 - 2)2 / 6
- Brackets first: 8 - 2 = 6
- Indices: 62 = 36
- Multiplication: 4 x 36 = 144
- Division: 144 / 6 = 24
- Addition: 3 + 24 = 27
Prime numbers and prime factorisation
A prime number has exactly two factors: 1 and itself. The number 1 is not prime (it has only one factor). The number 2 is the only even prime number.
Prime factorisation means expressing an integer as a product of its prime factors. The edexcel igcse mathematics specification a definition requires you to express integers as a product of powers of prime factors.
Worked example: Express 360 as a product of prime factors.
- 360 / 2 = 180
- 180 / 2 = 90
- 90 / 2 = 45
- 45 / 3 = 15
- 15 / 3 = 5
- 5 / 5 = 1
- 360 = 23 x 32 x 5
HCF and LCM using prime factorisation
Once you have the prime factorisation of two numbers, finding the highest common factor (HCF) and lowest common multiple (LCM) is systematic.
HCF: Take the lowest power of each prime factor that appears in both numbers.
LCM: Take the highest power of each prime factor that appears in either number.
Worked example: Find the HCF and LCM of 72 and 120.
- 72 = 23 x 32
- 120 = 23 x 3 x 5
- HCF = 23 x 3 = 24 (lowest power of each common prime)
- LCM = 23 x 32 x 5 = 360 (highest power of each prime from either)
You can verify: HCF x LCM = 24 x 360 = 8640 = 72 x 120. This check is a quick way to confirm your answer in the exam.
Exam question patterns for integers
The edexcel igcse mathematics specification a notes on integers show several recurring patterns in past papers.
Pattern 1: Ordering integers. Place integers on a number line or order a set from smallest to largest. Remember that -5 is less than -2, because it sits further left on the number line.
Pattern 2: Identifying factors, multiples or primes from a list. "From the numbers 12, 13, 14, 15, 16, write down a prime number." Answer: 13. Candidates lose this mark by choosing 15 (divisible by 3 and 5).
Pattern 3: Using BIDMAS in context. "A mobile phone plan charges 10 + 0.05 x (minutes - 100) for any minutes over 100. Calculate the cost for 250 minutes." The brackets, multiplication and addition must be processed in the correct order.
Pattern 4: Prime factorisation to find HCF/LCM. Often worth 3-4 marks. Show the factor tree or repeated division, write the final answer in index form, then apply the HCF/LCM rules.
Common mistakes
- Listing 1 as a prime number. It is not. Primes have exactly two factors.
- Sign errors in directed number calculations. Write the sign explicitly at every step. Do not try to carry signs mentally.
- Ignoring BIDMAS. Calculate multiplication and division before addition and subtraction, always.
- Incomplete prime factorisation. Stopping at 4 x 15 instead of continuing to 22 x 3 x 5. Every factor must be prime.
Applying integers to real-world problems
The specification requires you to use directed numbers in practical situations. The exam frequently wraps integer arithmetic into contexts like temperature, altitude, bank balances and floor levels in a building.
Worked example: A submarine is at a depth of -120 m. It rises 45 m, then dives 30 m. What is its final depth?
- Start: -120
- Rise 45: -120 + 45 = -75
- Dive 30: -75 - 30 = -105
- Final depth: -105 m
The key is to interpret "rise" as adding a positive value and "dive" as subtracting. If you track the sign at each step rather than trying to do the whole calculation at once, the risk of error drops sharply.
Worked example: The temperature in Moscow at midnight was -14 degrees C. By noon, it had risen by 9 degrees C. By midnight the following day, it had fallen by 12 degrees C from the noon temperature. Find the final temperature.
- Midnight: -14
- Noon: -14 + 9 = -5
- Next midnight: -5 - 12 = -17 degrees C
Divisibility rules and their use
Knowing the basic divisibility rules speeds up factor-finding and prime factorisation.
| Divisible by | Rule |
|---|---|
| 2 | Last digit is even (0, 2, 4, 6, 8) |
| 3 | Sum of digits is divisible by 3 |
| 4 | Last two digits form a number divisible by 4 |
| 5 | Last digit is 0 or 5 |
| 6 | Divisible by both 2 and 3 |
| 9 | Sum of digits is divisible by 9 |
| 10 | Last digit is 0 |
These rules are not directly tested, but they save time when decomposing large numbers into prime factors. For instance, to factorise 882: it is even (divisible by 2), giving 441. Is 441 divisible by 3? The digit sum is 4 + 4 + 1 = 9, which is divisible by 3, giving 147. The digit sum of 147 is 12, divisible by 3, giving 49 = 72. So 882 = 2 x 32 x 72.
Being fluent with these small efficiencies across the Edexcel IGCSE Mathematics Specification A exam adds up to significant time savings, especially when HCF and LCM questions require clean, rapid factorisation.
Integers in algebra and beyond
Integer skills are not confined to the "Numbers and the number system" section of the specification. They reappear throughout the course. Collecting like terms in algebra requires adding and subtracting integers (including negatives). Substituting values into formulae means correctly handling BIDMAS with mixed signs. Solving equations involves inverse operations on integers. Index laws for integer powers use the same multiplication and division rules. If your integer arithmetic is unreliable, those errors propagate into every other topic, costing marks far beyond the number questions themselves. Investing time in mastering integer operations produces returns across the entire Edexcel IGCSE Mathematics Specification A course.
Self-check questions
- Calculate -8 + 3 x (-2)
- List all prime numbers between 20 and 40.
- Express 540 as a product of prime factors in index form.
- Find the HCF and LCM of 48 and 180.
- A temperature drops from 5 degrees C to -7 degrees C. What is the change in temperature?
Practise integers questions on the Green Bridge CBT platform using Edexcel IGCSE Mathematics Specification A past questions to build fluency. The edexcel igcse mathematics specification a explained answers show full working for every step.
Integers explained for the edexcel igcse mathematics specification a exam: place value, directed numbers, factors, multiples, primes and worked examples.
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