Types of Number, Factors & Powers

Mathematics· 1.1, 1.3· 15 min read

1. Classifying Common Number Types★☆☆☆☆⏱ 4 min

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.

📘 Definition

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
1. 11 is only divisible by 1 and itself → prime number
3
1. √3 is a non-terminating, non-recurring decimal → irrational number
4
1. 5/8 is a fraction of two integers → rational number
5
1. 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.

2. Prime Factorisation of Positive Integers★★☆☆☆⏱ 4 min

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.

📘 Definition

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

3. Calculating HCF and LCM★★★☆☆⏱ 4 min

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

📐 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
1. For HCF, take the lowest power of each common prime: 2² × 3¹ × 5¹ = 4 × 3 × 5 = 60
3
1. 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.

4. Recall of Squares, Cubes and Roots★☆☆☆☆⏱ 3 min

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
1. 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.

5. Common Pitfalls

Wrong move:

Including 1 as a prime number

Why:

1 has only one factor, so it is not classified as prime, leading to incorrect prime factorisation

Correct move:

Start prime factorisation with 2, the smallest prime number, and exclude 1 from all prime factor lists

Wrong move:

Swapping HCF and LCM power rules

Why:

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 move:

Use the mnemonic 'HCF-Low, LCM-High' to apply the correct power rules

Wrong move:

Classifying recurring decimals as irrational

Why:

All recurring or terminating decimals can be written as a fraction of two integers, so they are rational

Correct move:

Only classify non-terminating, non-recurring decimals (e.g. π, √2) as irrational

Wrong move:

Omitting primes when calculating LCM

Why:

Leaving out a prime that appears in only one factorisation gives an LCM that is not a multiple of that number

Correct move:

List all primes present across all factorisations before taking the highest power of each

Wrong move:

Writing square roots as both positive and negative in basic number questions

Why:

Core IGCSE uses the radical symbol to denote the positive principal root unless specified otherwise

Correct move:

Only give the positive root for questions asking for √n unless you are solving a quadratic equation

6. Quick Reference 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

7. Frequently Asked

Do I need to memorise squares and cubes for Core papers?

Yes, CIE IGCSE 0580 Core expects you to recall squares of 1-15 and cubes of 1-5 and 10 without calculation to save time in non-calculator papers.

What is the difference between HCF and LCM?

HCF is the largest number that divides all given values evenly, while LCM is the smallest number that all given values divide into evenly. Both are calculated using prime factorisation.

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.