# Integers

> Edexcel International GCSE Mathematics A · 4MA1 (2016 Spec)
> Source: https://www.owlsprep.com/study/edexcel-igcse-math-a-s1-integers/

This guide covers all integer content for Edexcel IGCSE Maths A (4MA1) spec point 1.1, including directed number operations, BIDMAS, primes, factors, multiples and ordering rules for all exam papers.

**Prerequisites:** Basic number recognition and counting skills

## Learning objectives

- Recognise and order positive, negative integers and zero, and apply place value rules correctly
- Perform addition, subtraction, multiplication and division with directed integers in practical contexts
- Follow BIDMAS hierarchy to solve multi-step integer calculations including brackets
- Define and identify odd, even, prime numbers, factors and multiples of integers
- Find prime factors, common factors and common multiples of sets of integers

## Core Integer Definitions and Ordering

**Integer** — Whole numbers including positive values, negative values and zero, with no fractional or decimal components.

*Example:* $-5, 0, 3, 172$ are integers; $0.6, \frac{1}{2}, -3.2$ are not integers.

Integers follow standard place value rules, where each digit's value depends on its position (units, tens, hundreds, thousands etc.). When ordering integers: negative numbers are smaller the further they are from zero, while positive numbers are larger the further they are from zero.

**Worked example:** Order the following integers from smallest to largest: $7, -2, 0, -9, 3$

1. 1. Separate values into negative integers, zero and positive integers: negatives = $[-2, -9]$, zero = $0$, positives = $[7, 3]$
2. 2. Order negative values from smallest to largest: more negative values come first, so $-9 < -2$
3. 3. Order positive values from smallest to largest: $3 < 7$
4. 4. Combine the lists in order: $-9, -2, 0, 3, 7$

> **Exam tip:** Double-check the order of negative integers: a very common error is writing $-2$ before $-9$ when sorting from smallest to largest.

## Operations with Directed Integers

**Directed Number** — Numbers with a positive or negative sign that indicate direction relative to zero, often used in practical contexts like temperatures, elevation or bank balances.

- Adding a negative number = subtracting a positive number: $3 + (-2) = 3 - 2 = 1$
- Subtracting a negative number = adding a positive number: $5 - (-3) = 5 + 3 = 8$
- Multiplying/dividing two numbers with the same sign gives a positive result
- Multiplying/dividing two numbers with different signs gives a negative result

**Worked example:** The temperature in Oslo at midnight is $-4^\circ C$. By midday, the temperature rises by $9^\circ C$, then drops by $12^\circ C$ by 6pm. What is the temperature at 6pm?

1. 1. Start with the initial midnight temperature: $-4^\circ C$
2. 2. Add the temperature rise at midday: $-4 + 9 = 5^\circ C$
3. 3. Subtract the temperature drop by 6pm: $5 - 12 = -7^\circ C$
4. Final answer: $-7^\circ C$

> **tip**
>
> Always include units (e.g. $^\circ C$, £) in your final answer for practical directed number questions to avoid losing easy marks.

## Order of Operations (BIDMAS)

**BIDMAS** — Standard hierarchy of operations for multi-step calculations: Brackets first, then Indices, then Division and Multiplication (equal priority, left to right), then Addition and Subtraction (equal priority, left to right).

**Worked example:** Calculate the value of $12 - 3 \times (4 + 2) \div 2$

1. 1. First calculate the value inside the brackets: $4 + 2 = 6$
2. 2. Simplify the expression to: $12 - 3 \times 6 \div 2$
3. 3. Complete multiplication and division left to right: $3 \times 6 = 18$, then $18 \div 2 = 9$
4. 4. Complete the final subtraction: $12 - 9 = 3$
5. Final answer: $3$

> **Exam tip:** Never calculate left to right without following BIDMAS: this is one of the most common sources of avoidable mark loss in arithmetic questions.

## Prime Numbers, Factors and Multiples

- **Even number**: Integer divisible by 2 (ends in 0, 2, 4, 6, 8)
- **Odd number**: Integer not divisible by 2 (ends in 1, 3, 5, 7, 9)
- **Prime number**: Positive integer with exactly two distinct positive factors (1 and itself; 1 is not prime)
- **Factor**: Integer that divides another integer exactly with no remainder
- **Multiple**: Product of an integer and any positive whole number

**Worked example:** List the prime factors, common factors and first 3 common multiples of 12 and 18.

1. 1. Prime factors of 12: 2, 3; Prime factors of 18: 2, 3
2. 2. List all factors of 12: 1, 2, 3, 4, 6, 12; All factors of 18: 1, 2, 3, 6, 9, 18
3. 3. Common factors are values in both lists: 1, 2, 3, 6
4. 4. List multiples of 12: 12, 24, 36, 48, 60, 72...; Multiples of 18: 18, 36, 54, 72, 90...
5. 5. First 3 common multiples: 36, 72, 108

**Check your understanding**

1. Which of the following is a prime number?

   - A) 1
   - B) 9
   - C) 13
   - D) 21

   *Why:* 13 only has two factors: 1 and 13. 1 has only one factor, 9 has factors 1, 3, 9, and 21 has factors 1, 3, 7, 21.

## Common pitfalls

- **Wrong:** Ordering integers from smallest to largest as $-2, -9, 0, 3, 7$
  - Why it fails: Assuming larger digit values mean larger numbers for negative integers
  - Correct: More negative numbers are smaller, so the correct order is $-9, -2, 0, 3, 7$
- **Wrong:** Calculating $5 - (-3) = 2$
  - Why it fails: Treating subtraction of a negative as subtraction of a positive
  - Correct: Subtracting a negative equals adding a positive, so $5 + 3 = 8$
- **Wrong:** Classifying 1 as a prime number
  - Why it fails: Misremembering the prime number definition as 'number only divisible by 1 and itself' without the 'exactly two distinct factors' requirement
  - Correct: 1 only has one factor (itself), so it is not a prime number
- **Wrong:** Calculating $2 + 3 \times 4 = 20$ by working left to right
  - Why it fails: Ignoring BIDMAS order of operations
  - Correct: Complete multiplication first: $3 \times 4 = 12$, then add 2 to get 14
- **Wrong:** Calculating $(-4) \times (-5) = -20$
  - Why it fails: Misapplying sign rules for multiplication of directed numbers
  - Correct: Two negative numbers multiplied give a positive result, so the answer is 20

## Cheatsheet

| Concept | Rule/Definition |
| --- | --- |
| Integer | Whole number (positive, negative, zero; no fractions/decimals) |
| ×/÷ Sign Rules | Same sign = positive, Different sign = negative |
| BIDMAS Order | Brackets > Indices > ×/÷ (left-right) > +/− (left-right) |
| Prime Number | Exactly 2 factors: 1 and itself; 1 is not prime |
| Factor | Divides a number exactly with no remainder |
| Multiple | Product of a number and any positive integer |

## What's next

Integers are the foundation of all Edexcel IGCSE Maths A content, so mastering these rules is critical for success in every other topic. You can now move on to learning about powers and roots, where you will apply BIDMAS and prime factor knowledge to solve more complex arithmetic problems, as specified in syllabus point 1.4. You can also practice integer calculation questions to build speed and accuracy for your exam, and review other number system topics such as fractions and decimals to strengthen your core arithmetic skills.

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