What you'll learn
This topic covers the different types of number you meet throughout maths — integers, fractions, decimals, primes, factors and multiples — and the tools for working with them. In this guide you will learn how to classify numbers, find factors and multiples, identify prime numbers, write a number as a product of its prime factors, and use the HCF and LCM. These number skills underpin almost every other topic.
Key terms and definitions
Integer — a whole number, positive, negative or zero (… −2, −1, 0, 1, 2 …).
Factor — a number that divides exactly into another with no remainder.
Multiple — the result of multiplying a number by an integer.
Prime number — a number with exactly two factors: 1 and itself.
HCF — highest common factor of two or more numbers.
LCM — lowest common multiple of two or more numbers.
Core concepts
Classifying numbers
Numbers come in types: integers (whole numbers), fractions (parts of a whole, e.g. ¾), decimals (e.g. 0.75) and rational numbers (any number that can be written as a fraction). A fraction, its equivalent decimal and percentage all describe the same value. Knowing the type helps you choose the right method.
Factors and multiples
A factor divides exactly into a number (factors of 12: 1, 2, 3, 4, 6, 12). A multiple is what you get when you multiply by an integer (multiples of 5: 5, 10, 15, 20 …). Every number is a factor of its own multiples. Listing factors in pairs (1 × 12, 2 × 6, 3 × 4) ensures you find them all.
Prime numbers
A prime number has exactly two factors, 1 and itself: 2, 3, 5, 7, 11, 13, 17, 19, 23 … Note that 1 is not prime (it has only one factor) and 2 is the only even prime. Primes are the building blocks of all whole numbers.
Prime factorisation
Every integer can be written as a product of prime factors. Use a factor tree, splitting the number into factors repeatedly until all branches are prime, then write the answer with powers. For example, 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5.
HCF and LCM
The HCF (highest common factor) is the largest number dividing into all the given numbers; the LCM (lowest common multiple) is the smallest number they all divide into. Using prime factors: the HCF multiplies the shared primes (lowest powers), and the LCM multiplies all primes (highest powers). A Venn diagram of prime factors makes this clear.
Worked examples
Example 1: Prime factorisation
Write 84 as a product of prime factors.
84 = 2 × 42 = 2 × 2 × 21 = 2 × 2 × 3 × 7 = 2² × 3 × 7.
Example 2: HCF
Find the HCF of 24 and 36.
24 = 2³ × 3, 36 = 2² × 3². Shared: 2² × 3 = 12.
Example 3: LCM
Find the LCM of 6 and 8.
6 = 2 × 3, 8 = 2³. LCM = 2³ × 3 = 24.
Common mistakes and how to avoid them
Calling 1 prime. A prime has exactly two factors; 1 has only one.
Confusing factors and multiples. Factors divide into the number; multiples are bigger (times tables).
Mixing up HCF and LCM. HCF is the largest that divides in; LCM is the smallest they go into.
Missing factors. List in pairs to be sure you have them all.
Stopping the factor tree early. Keep splitting until every branch is prime.
Exam technique for Types of Number
Know the definitions of integer, factor, multiple and prime.
List factors in pairs so none are missed.
Use a factor tree for prime factorisation, then write with powers.
Use prime factors to find HCF (shared, lowest powers) and LCM (all, highest powers).
Read the question — does it want the highest common or the lowest common value?
Quick revision summary
Numbers are classified as integers (whole numbers), fractions, decimals and rational numbers, with the same value expressible in different forms. A factor divides exactly into a number (list in pairs to find them all); a multiple is the number times an integer. A prime number has exactly two factors — 2, 3, 5, 7, 11 … — with 1 not prime and 2 the only even prime. Every integer is a product of prime factors, found with a factor tree and written using powers (60 = 2² × 3 × 5). The HCF is the largest number dividing into all the given numbers (multiply shared primes, lowest powers) and the LCM is the smallest they all divide into (multiply all primes, highest powers); a prime-factor Venn diagram helps. The common errors are calling 1 prime, swapping factors and multiples, mixing up HCF and LCM, and stopping the factor tree too soon. Learn the definitions, list factors in pairs, use a factor tree, and apply prime factors to HCF and LCM.