Number

Factors, Multiples, Primes, HCF & LCM

Year 7 · Year 8 · Year 9

✓By the end of this lesson students will be able to identify factors and multiples of a given integer.
✓By the end of this lesson students will be able to recognise prime numbers and composite numbers.
✓By the end of this lesson students will be able to express a number as a product of its prime factors using a factor tree.
✓By the end of this lesson students will be able to find the Highest Common Factor (HCF) of two or more numbers.
✓By the end of this lesson students will be able to find the Lowest Common Multiple (LCM) of two or more numbers.

Key concepts

Factors

Factors are whole numbers that divide exactly into another number without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.

Multiples

Multiples are the results of multiplying a number by an integer. For example, the multiples of 5 are 5, 10, 15, 20, and so on.

Prime Numbers

A prime number is a whole number greater than 1 that has exactly two factors: 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, ...

Prime Factorisation

Prime factorisation is the process of writing a number as a product of its prime factors. This representation is unique for every composite number. A common method is using a factor tree.

Highest Common Factor (HCF)

The Highest Common Factor (HCF) of two or more numbers is the largest number that is a factor of all the given numbers. It can be found by identifying the common prime factors from their prime factorisations and multiplying them together.

Lowest Common Multiple (LCM)

The Lowest Common Multiple (LCM) of two or more numbers is the smallest positive number that is a multiple of all the given numbers. It can be found by multiplying all the prime factors from the prime factorisations of the numbers, ensuring each common factor is only counted once.

Key facts to remember

11 is not a prime number.
22 is the only even prime number.
3Every composite number can be expressed as a unique product of prime factors.
4A factor tree is a systematic way to find the prime factorisation of a number.
5The HCF is found by multiplying the prime factors that are common to all numbers.
6The LCM is found by multiplying all the prime factors from the numbers' prime factorisations, ensuring common factors are only included once.
7For any two numbers A and B, HCF(A,B) × LCM(A,B) = A × B.

Worked examples

Example 1

Express 72 as a product of its prime factors.

IStart with 72 and find any two factors, e.g., 2 and 36.

II72 = 2 × 36

III2 is prime, so circle it. Now break down 36.

IV36 = 2 × 18

V2 is prime, circle it. Now break down 18.

VI18 = 2 × 9

VII2 is prime, circle it. Now break down 9.

VIII9 = 3 × 3

9Both 3s are prime, circle them.

10Collect all the circled prime factors: 2, 2, 2, 3, 3.

Answer

72 = 2 × 2 × 2 × 3 × 3 or 2³ × 3²

A factor tree is a visual way to do this. Always break down numbers until all branches end in a prime number.

Example 2

Find the HCF and LCM of 60 and 84.

IStep 1: Find the prime factorisation of each number.

IIFor 60:

III60 = 2 × 30

IV30 = 2 × 15

V15 = 3 × 5

VISo, 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5

VIIFor 84:

VIII84 = 2 × 42

942 = 2 × 21

1021 = 3 × 7

11So, 84 = 2 × 2 × 3 × 7 = 2² × 3 × 7

12Step 2: Use a Venn diagram to organise the prime factors.

13Common factors (intersection): 2, 2, 3

14Factors unique to 60: 5

15Factors unique to 84: 7

16Step 3: Calculate the HCF.

17HCF = product of common prime factors (intersection)

18HCF = 2 × 2 × 3 = 12

19Step 4: Calculate the LCM.

20LCM = product of all prime factors (union)

21LCM = (2 × 2 × 3) × 5 × 7 = 12 × 5 × 7 = 60 × 7 = 420

Answer

HCF = 12, LCM = 420

The Venn diagram method clearly shows common factors for HCF and all factors for LCM.

Example 3

Find the HCF and LCM of 48 and 72.

IStep 1: Find the prime factorisation of each number.

IIFor 48:

III48 = 2 × 24

IV24 = 2 × 12

V12 = 2 × 6

VI6 = 2 × 3

VIISo, 48 = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3

VIIIFor 72:

972 = 2 × 36

1036 = 2 × 18

1118 = 2 × 9

129 = 3 × 3

13So, 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3²

14Step 2: Identify common and unique prime factors.

15Common factors: three 2s and one 3 (2 × 2 × 2 × 3)

16Unique to 48: one 2

17Unique to 72: one 3

18Step 3: Calculate the HCF.

19HCF = product of common prime factors

20HCF = 2 × 2 × 2 × 3 = 24

21Step 4: Calculate the LCM.

22LCM = product of all prime factors (common and unique)

23LCM = (2 × 2 × 2 × 3) × 2 × 3 = 24 × 2 × 3 = 48 × 3 = 144

Answer

HCF = 24, LCM = 144

You can also use the formula HCF(A,B) × LCM(A,B) = A × B to check your answer. 24 × 144 = 3456. 48 × 72 = 3456.

Common mistakes

✗Including 1 as a prime factor in prime factorisation.
✗Confusing the definitions of factors and multiples.
✗Not breaking down numbers completely into prime factors in a factor tree (e.g., stopping at 4 × 9 instead of 2 × 2 × 3 × 3).
✗Incorrectly identifying common factors or multiples, especially when using Venn diagrams for HCF and LCM.
✗Forgetting to include the unique prime factors when calculating the LCM, or including common factors multiple times unnecessarily.

Exam tips

★Always show your working, especially factor trees or Venn diagrams, as method marks are often awarded.
★Double-check your prime factorisation by multiplying the prime factors together to ensure they equal the original number.
★Use a Venn diagram to clearly organise the prime factors when finding HCF and LCM of two or more numbers.
★Read the question carefully to determine whether you need to find the HCF or the LCM, as they are distinct concepts.

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