Factors, Multiples and Primes

This lesson covers factors, multiples, prime numbers, prime factorisation, and finding the Highest Common Factor (HCF) and Lowest Common Multiple (LCM). These are fundamental Number topics in the Edexcel GCSE Mathematics (1MA1) specification, tested on all tiers and frequently appearing on Paper 1 (non-calculator).

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Key Vocabulary

Term	Meaning
Factor	A number that divides exactly into another number
Multiple	The result of multiplying a number by an integer
Prime number	A number with exactly two factors: 1 and itself
Composite number	A number with more than two factors
Prime factor	A factor that is a prime number
Prime factorisation	Writing a number as a product of its prime factors
HCF	Highest Common Factor — the largest factor shared by two or more numbers
LCM	Lowest Common Multiple — the smallest multiple shared by two or more numbers

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Factors

The factors of a number are all the whole numbers that divide into it exactly.

Worked Example 1

Find all the factors of 36.

Work systematically in pairs:

• $1 \times 36$1×36
• $2 \times 18$2×18
• $3 \times 12$3×12
• $4 \times 9$4×9
• $6 \times 6$6×6

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

Tip: Always start from 1 and work upwards. Stop when the factor pairs start repeating.

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Multiples

The multiples of a number are found by multiplying it by 1, 2, 3, 4, ...

Worked Example 2

List the first six 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.

The first 15 prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Important points:

• 1 is NOT a prime number (it has only one factor).
• 2 is the only even prime number.
• Every whole number greater than 1 is either prime or can be written as a product of primes.

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Prime Factorisation

Every composite number can be written as a product of its prime factors. This is called prime factorisation.

Method 1: Factor Tree

Worked Example 3

Write 180 as a product of its prime factors.

        180
       /   \
      10    18
     / \   / \
    2   5  2   9
              / \
             3   3

Collecting the prime factors from the ends of the branches:

$180 = 2 \times 5 \times 2 \times 3 \times 3 = 2^{2} \times 3^{2} \times 5$180=2×5×2×3×3=22×32×5

Answer: $180 = 2^{2} \times 3^{2} \times 5$180=22×32×5

Method 2: Repeated Division

Divide by the smallest prime factor repeatedly:

Division	Result
$180 \div 2$180÷2	90
$90 \div 2$90÷2	45
$45 \div 3$45÷3	15
$15 \div 3$15÷3	5
$5 \div 5$5÷5	1

$180 = 2 \times 2 \times 3 \times 3 \times 5 = 2^{2} \times 3^{2} \times 5$180=2×2×3×3×5=22×32×5 ✓

Edexcel Exam Tip: Always write your final answer using index notation (e.g. $2^{2} \times 3^{2} \times 5)$22×32×5). Edexcel mark schemes specifically award marks for the index form. Show your factor tree or repeated division to earn full method marks.

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Highest Common Factor (HCF)

The HCF of two or more numbers is the largest number that is a factor of all of them.

Method 1: Listing Factors

Worked Example 4

Find the HCF of 24 and 36.

• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
• Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
• Common factors: 1, 2, 3, 4, 6, 12

HCF = 12

Method 2: Prime Factorisation

Worked Example 5

Find the HCF of 60 and 84.

• $60 = 2^{2} \times 3 \times 5$60=22×3×5
• $84 = 2^{2} \times 3 \times 7$84=22×3×7

For HCF, take the lowest power of each common prime factor:

HCF $= 2^{2} \times 3 = 4 \times 3 =$=22×3=4×3= 12

Method 3: Venn Diagram

Worked Example 6

Find the HCF of 120 and 150 using a Venn diagram.

• $120 = 2^{3} \times 3 \times 5$120=23×3×5
• $150 = 2 \times 3 \times 5^{2}$150=2×3×52

Draw a Venn diagram with the prime factors:

    120 only    |   Both   |  150 only
      2, 2      |  2, 3, 5 |     5

HCF = product of factors in the overlap $= 2 \times 3 \times 5 =$=2×3×5= 30

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Lowest Common Multiple (LCM)

The LCM of two or more numbers is the smallest number that is a multiple of all of them.

Method 1: Listing Multiples

Worked Example 7

Find the LCM of 6 and 8.

• Multiples of 6: 6, 12, 18, 24, 30, ...
• Multiples of 8: 8, 16, 24, 32, ...

LCM = 24

Method 2: Prime Factorisation

Worked Example 8

Find the LCM of 60 and 84.

• $60 = 2^{2} \times 3 \times 5$60=22×3×5
• $84 = 2^{2} \times 3 \times 7$84=22×3×7

For LCM, take the highest power of each prime factor that appears in either number:

LCM $= 2^{2} \times 3 \times 5 \times 7 = 4 \times 3 \times 5 \times 7 =$=22×3×5×7=4×3×5×7= 420

Method 3: Venn Diagram

Using the Venn diagram from Worked Example 6:

    120 only    |   Both   |  150 only
      2, 2      |  2, 3, 5 |     5

LCM = product of ALL factors in the Venn diagram $= 2 \times 2 \times 2 \times 3 \times 5 \times 5 =$=2×2×2×3×5×5= 600

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Useful Relationship: HCF $\times$× LCM = Product of the Two Numbers

For any two numbers a and b:

HCF(a, $b) \times$b)× LCM(a, $b) = a \times b$b)=a×b

Worked Example 9

The HCF of two numbers is 6 and their LCM is 120. One number is 24. Find the other number.

HCF $\times$× LCM = product of the two numbers

$6 \times 120 = 24 \times$6×120=24× other number

$720 = 24 \times$720=24× other number

Other number $= 720 \div 24 =$=720÷24= 30

Check: HCF(24, 30) = 6 ✓, LCM(24, 30) = 120 ✓

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Applications of HCF and LCM

Worked Example 10

Buses leave a station every 12 minutes and trains leave every 20 minutes. A bus and a train leave together at 09:00. When will they next leave together?

Find LCM(12, 20):

• $12 = 2^{2} \times 3$12=22×3
• $20 = 2^{2} \times 5$20=22×5
• LCM $= 2^{2} \times 3 \times 5 = 60$=22×3×5=60 minutes

Answer: They next leave together at 10:00 (60 minutes later).

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Common Mistakes and Misconceptions

Mistake	Correction
Thinking 1 is prime	1 is NOT prime — it has only one factor
Confusing HCF and LCM	HCF is always $\leq$≤ both numbers; LCM is always $\geq$≥ both numbers
Not using index notation for prime factorisation	Always write $2^{3} \times 3 \times 5$23×3×5, not $2 \times 2 \times 2 \times 3 \times 5$2×2×2×3×5
For HCF, taking the highest powers	Take the LOWEST powers of COMMON primes
For LCM, taking only common primes	Take the HIGHEST powers of ALL primes

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Practice Problems

1. Write 360 as a product of its prime factors.
2. Find the HCF of 48 and 72.
3. Find the LCM of 15 and 20.
4. Use prime factorisation to find the HCF and LCM of 126 and 168.
5. Two lights flash. One flashes every 8 seconds, the other every 14 seconds. They flash together at time 0. When do they next flash together?
6. The HCF of two numbers is 8 and their LCM is 96. One number is 32. Find the other.
7. Explain why 2 is the only even prime number.
8. Find the HCF and LCM of 36, 48 and 60.

Answers