Factors, Multiples & Primes

Spec reference: Number - Structure and calculation
Key idea: Find factors, multiples and prime numbers; use prime factor decomposition to find HCF and LCM.

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Factors

A factor of a number divides into it exactly with no remainder.

Example

Find all factors of 24.

Work in factor pairs: $1\times 24,2\times 12,3\times 8,4\times 6$

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

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Multiples

A multiple of a number is in that number's times table.

Example

First 6 multiples of 7: 7, 14, 21, 28, 35, 42

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Prime numbers

A prime number has exactly two factors: 1 and itself.

First 10 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

Key facts

• 1 is not a prime number (only one factor)
• 2 is the only even prime number

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Prime factor decomposition

Every integer greater than 1 can be written as a product of prime factors.

Use a factor tree to break the number down:

Example

Write 60 as a product of its prime factors.

$$60=2\times 30=2\times 2\times 15=2\times 2\times 3\times 5$$$$\boxed{{\displaystyle 60=2^2\times 3\times 5}}$$

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Highest Common Factor (HCF)

The HCF is the largest number that divides into both numbers exactly.

Example

Find the HCF of 24 and 36.

$$24=2^3\times 3$$$$36=2^2\times 3^2$$

Take the lowest power of each common prime factor:

$$\text{HCF}=2^2\times 3=4\times 3=\boxed{{\displaystyle 12}}$$

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Lowest Common Multiple (LCM)

The LCM is the smallest number that is a multiple of both numbers.

Example

Find the LCM of 8 and 12.

$$8=2^3$$$$12=2^2\times 3$$

Take the highest power of each prime factor:

$$\text{LCM}=2^3\times 3=8\times 3=\boxed{{\displaystyle 24}}$$

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Venn diagram method

You can use a Venn diagram to find HCF and LCM at the same time:

• Left circle: prime factors of first number only
• Overlap: prime factors in both numbers
• Right circle: prime factors of second number only

$$\text{HCF}=\text{product of the overlap}$$$$\text{LCM}=\text{product of everything in both circles}$$

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Exam tips

Watch out for

• Always check factor trees end in prime numbers
• LCM is for "at the same time again" problems (e.g. buses, lights flashing)
• HCF is for "sharing equally" or "cutting into equal pieces" problems

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Test Yourself

Question 1 of 5

What is the HCF of 36 and 48?