Multiples, Factors and Prime Factors Revision Notes for Edexcel GCSE Maths

Multiples, factors and prime factors

Understanding multiples, factors and prime factors is fundamental to working with numbers in mathematics. These concepts help us break down numbers and understand their relationships with each other.

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These three concepts are interconnected - multiples help us understand how numbers relate in sequences, factors show us how to numbers can be divided, and prime factors reveal the fundamental building blocks of all numbers.

What are multiples?

Multiples of a number are simply the results you get from its times table. When you multiply a number by 1, 2, 3, 4, and so on, you create a sequence of multiples.

To find multiples of any number, you just need to work through its times table systematically. The key function is to multiply the number by consecutive positive integers.

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Worked Example: Finding the First 8 Multiples of 13

To find the first 8 multiples of 13, we calculate:

$13 \times 1 = 13$
$13 \times 2 = 26$
$13 \times 3 = 39$
$13 \times 4 = 52$
$13 \times 5 = 65$
$13 \times 6 = 78$
$13 \times 7 = 91$
$13 \times 8 = 104$

Therefore, the first 8 multiples of 13 are: 13, 26, 39, 52, 65, 78, 91, 104.

Understanding factors

Factors of a number are all the whole numbers that divide into it exactly, leaving no remainder. Finding factors is like discovering all the different ways you can multiply two numbers together to get your original number.

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Systematic Method for Finding All Factors:

Start with $1 \times \text{the number itself}$ , then try $2 \times$ , then $3 \times$ and so on, listing the pairs in rows
Test each number in turn - if it doesn't divide exactly, cross out that row
When you reach a number that repeats from earlier calculations, stop
The numbers in the rows you haven't crossed out give you the complete list of factors

This systematic approach is essential for ensuring you don't miss any factors and avoid duplicates.

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Worked Example: Finding All Factors of 24

To find all factors of 24, we test each number systematically:

$1 \times 24 = 24$ ✓
$2 \times 12 = 24$ ✓
$3 \times 8 = 24$ ✓
$4 \times 6 = 24$ ✓
$5$ doesn't divide 24 exactly ✗
$6 \times 4 = 24$ (already found this pair)

Therefore, the factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24.

Prime factorization using factor trees

Any number can be broken down into prime numbers multiplied together. This process is called expressing a number as a product of its prime factors, and it's like finding the basic building blocks of that number.

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Factor Tree Method Steps:

Start with your number at the top and split it into any two factors
Every time you get a prime number, circle it - these are your final answers
Keep splitting the non-prime numbers until you can't go any further
When you're left with only prime numbers, write them out in order

The factor tree method is an organised way to find prime factors and provides a visual representation of the breakdown process.

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Worked Example: Prime Factorization of 420

Using the factor tree method for 420:

$420 = 42 \times 10$
$42 = 6 \times 7$ (7 is prime, so circle it)
$10 = 2 \times 5$ (both prime, so circle them)
$6 = 2 \times 3$ (both prime, so circle them)

This gives us: $420 = 2 \times 2 \times 3 \times 5 \times 7 = 2^2 \times 3 \times 5 \times 7$

The factor tree helps you visualise this breakdown clearly and ensures you don't miss any prime factors.

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Every positive integer has a unique prime factorization - this is known as the Fundamental Theorem of Arithmetic. No matter which factors you choose to split first, you'll always end up with the same prime factors.

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Key Points to Remember:

Multiples are created by multiplying a number by 1, 2, 3, 4, etc. - essentially its times table
Factors are numbers that divide exactly into your original number with no remainder
Use systematic methods to find all factors - start with 1 and work upwards, testing each number
Factor trees help break numbers down into their prime factors by repeatedly splitting into smaller factors
Prime factorization shows every number as a unique product of prime numbers multiplied together

Multiples, Factors and Prime Factors (Edexcel GCSE Maths): Revision Notes