# Types of Number, Factors & Powers

> Mathematics · CIE IGCSE 0580 (2025-2027)
> Source: https://www.owlsprep.com/study/cie-0580-u1-types-of-number-factors-powers/

This guide covers all Core content for CIE IGCSE 0580 number types, prime factorisation, HCF, LCM and powers, aligned to the 2025-2027 syllabus with exam-specific worked examples.

**Prerequisites:** Basic addition, subtraction, multiplication and division of whole numbers

## Learning objectives

- Classify numbers as natural, integer, prime, square, cube, rational or irrational
- Express positive integers as products of prime factors in index form
- Calculate HCF and LCM of two or more numbers using prime factorisation
- Recall squares 1-15, cubes of 1-5 and 10, and their corresponding roots
- Identify reciprocals of integers and fractions

## Classifying Common Number Types

All numbers in Core IGCSE Maths fall into distinct categories based on their properties. Correct classification is required for 1-2 mark questions across all Core papers.

**Number Classification Rules** — Categorise numbers using their core properties: natural numbers are positive counting values, integers include negatives and zero, primes have exactly two factors, rationals can be written as fractions, irrationals cannot.

**Worked example:** Classify the following numbers into their most specific category: -5, 11, √3, 5/8, 144

1. 1. -5 is a negative whole number → integer
2. 2. 11 is only divisible by 1 and itself → prime number
3. 3. √3 is a non-terminating, non-recurring decimal → irrational number
4. 4. 5/8 is a fraction of two integers → rational number
5. 5. 144 = 12² → square number

> **Exam tip:** For classification questions, only state the most specific category unless explicitly asked to list all applicable types to avoid wasting time.

## Prime Factorisation of Positive Integers

Every positive integer can be written uniquely as a product of prime numbers, called its prime factorisation. This is the foundation for calculating HCF and LCM, tested in 2-3 mark questions.

**Prime Factorisation** — The process of breaking a number down into the set of prime numbers that multiply to give the original value, usually written in ascending index form.

**Worked example:** Express 240 as a product of prime factors in index form.

1. 1. Divide 240 by the smallest prime factor, 2: 240 = 2 × 120
2. 2. Divide 120 by 2 repeatedly: 240 = 2⁴ × 15
3. 3. Divide 15 by the next smallest prime, 3: 240 = 2⁴ × 3 × 5
4. 4. 5 is prime, so final form is 2⁴ × 3 × 5

> **Exam tip:** Always write prime factors in ascending order with indices to avoid losing presentation marks.

## Calculating HCF and LCM

HCF and LCM are used across number, algebra and ratio questions. You will always use prime factorisation to calculate them for Core exams.

> **HCF/LCM Memory Hook**
>
> HCF = Lowest common powers, LCM = Highest all powers

**Worked example:** Find the HCF and LCM of 240 and 180 using their prime factorisations.

1. 1. First write both prime factorisations: 240 = 2⁴ × 3 × 5, 180 = 2² × 3² × 5
2. 2. For HCF, take the lowest power of each common prime: 2² × 3¹ × 5¹ = 4 × 3 × 5 = 60
3. 3. For LCM, take the highest power of every prime present: 2⁴ × 3² × 5¹ = 16 × 9 × 5 = 720

> **Exam tip:** Check your LCM by confirming it is divisible by both original numbers to catch calculation errors.

## Recall of Squares, Cubes and Roots

Core IGCSE requires you to memorise specific squares and cubes to save time in non-calculator papers, and to identify square/cube numbers quickly.

- Squares 1 to 15: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
- Cubes: 1³=1, 2³=8, 3³=27, 4³=64, 5³=125, 10³=1000

**Worked example:** Without a calculator, find √196 and ∛64.

1. 1. Recall 14² = 14 × 14 = 196, so √196 = 14
2. 2. Recall 4³ = 4 × 4 × 4 = 64, so ∛64 = 4

> **Exam tip:** If you forget a square, calculate it quickly by expanding (10 + x)² or (20 - x)², but memorisation will save you valuable time in the exam.

## Common pitfalls

- **Wrong:** Including 1 as a prime number
  - Why it fails: 1 has only one factor, so it is not classified as prime, leading to incorrect prime factorisation
  - Correct: Start prime factorisation with 2, the smallest prime number, and exclude 1 from all prime factor lists
- **Wrong:** Swapping HCF and LCM power rules
  - Why it fails: Using highest powers for HCF gives a value that is too large, using lowest powers for LCM gives a value that is too small
  - Correct: Use the mnemonic 'HCF-Low, LCM-High' to apply the correct power rules
- **Wrong:** Classifying recurring decimals as irrational
  - Why it fails: All recurring or terminating decimals can be written as a fraction of two integers, so they are rational
  - Correct: Only classify non-terminating, non-recurring decimals (e.g. π, √2) as irrational
- **Wrong:** Omitting primes when calculating LCM
  - Why it fails: Leaving out a prime that appears in only one factorisation gives an LCM that is not a multiple of that number
  - Correct: List all primes present across all factorisations before taking the highest power of each
- **Wrong:** Writing square roots as both positive and negative in basic number questions
  - Why it fails: Core IGCSE uses the radical symbol to denote the positive principal root unless specified otherwise
  - Correct: Only give the positive root for questions asking for √n unless you are solving a quadratic equation

## Cheatsheet

| Concept | Key Rule/Value |
| --- | --- |
| Natural numbers | 1, 2, 3, ... (positive counting numbers) |
| Integers | ...-3, -2, -1, 0, 1, 2, 3... |
| Prime numbers | Divisible only by 1 and itself (e.g. 2, 3, 5, 7, 11) |
| Prime factorisation | Write as product of primes in ascending index form |
| HCF | Lowest power of all common prime factors |
| LCM | Highest power of all prime factors present |
| Required squares | 1² to 15² (1 to 225) |
| Required cubes | 1³ to 5³, 10³ (1, 8, 27, 64, 125, 1000) |
| Rational number | Can be written as a/b (a, b integers, b≠0) |
| Irrational number | Non-terminating, non-recurring decimal, cannot be written as a fraction |

## What's next

Now that you have mastered the foundational number types, factors and powers for CIE IGCSE 0580 Core, you are ready to move on to more advanced number topics that build on these skills. Prime factorisation, HCF and LCM are used extensively in fraction simplification, ratio problems, and algebraic factorisation, so practicing these core concepts will make future topics much easier. You should also reinforce your memorisation of required squares and cubes to ensure you can answer non-calculator questions quickly and accurately. This topic is tested in every Core paper, so make sure you complete past paper questions to reinforce your learning.

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