Factors, Multiples, Primes, HCF and LCM

This lesson covers factors, multiples and prime numbers, how to write a number as a product of its prime factors, and how to use prime factorisation to find the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of two or more numbers. You will also meet the useful identity that, for two numbers, $\text{HCF} \times \text{LCM} = \text{product of the two numbers}$HCF×LCM=product of the two numbers. These ideas are part of the Number content of OCR GCSE Mathematics (J560) and appear regularly on the non-calculator Paper 1.

This lesson mainly builds AO1 fluency in factorising and listing, with AO2 reasoning when you justify whether a number is prime, and AO3 problem-solving in the worded HCF/LCM examples (for instance, when two events coincide).

Key Vocabulary

Term	Meaning
Factor	A whole number that divides exactly into another
Multiple	The result of multiplying a number by an integer
Prime number	A number with exactly two factors: $1$1 and itself
Composite number	A number with more than two factors
Prime factorisation	Writing a number as a product of prime numbers
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
Index form	Repeated factors written using powers, e.g. $2^{3} \times 5$23×5

Factors, Multiples and Primes

The factors of a number are all the whole numbers that divide into it with no remainder. The multiples of a number are what you get by multiplying it by $1, 2, 3, \ldots$1,2,3,… A prime number has exactly two factors — itself and $1$1 — so $1$1 is not prime (it has only one factor) and $2$2 is the only even prime.

Worked Example 1

Write down all the factors of $36$36.

Find factors in pairs that multiply to $36$36: $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.

Answer: $1, 2, 3, 4, 6, 9, 12, 18, 36$1,2,3,4,6,9,12,18,36 (nine factors). Because $36 = 6 \times 6$36=6×6 is a square number, the factor $6$6 appears only once.

Common error: Forgetting a pair such as $4 \times 9$4×9. Working in pairs, starting from $1$1 and moving up, stops you missing any.

Worked Example 2

Write down the first five multiples of $7$7, and state whether $7$7 is prime.

Multiples: $7, 14, 21, 28, 35$7,14,21,28,35. The number $7$7 has only the factors $1$1 and $7$7, so it is prime.

Prime Factorisation (Product of Primes)

Every integer greater than $1$1 can be written as a product of primes in exactly one way (ignoring order). A factor tree is the quickest way to find it: split the number into any factor pair, then keep splitting until every branch ends in a prime, and finally write the result in index form.

Worked Example 3

Write $360$360 as a product of its prime factors.

Build a factor tree. Below, circled-equivalent leaves are the primes.

graph TD
  A[360] --> B[36]
  A --> C[10]
  B --> D[6]
  B --> E[6]
  D --> F[2]
  D --> G[3]
  E --> H[2]
  E --> I[3]
  C --> J[2]
  C --> K[5]

Collect the prime leaves: $2, 3, 2, 3, 2, 5$2,3,2,3,2,5. In index form:

$360 = 2^{3} \times 3^{2} \times 5.$360=23×32×5.

Check: $2^{3} \times 3^{2} \times 5 = 8 \times 9 \times 5 = 360$23×32×5=8×9×5=360 ✓.

Common error: Stopping too early and leaving a composite leaf such as $6$6 in the answer. Keep splitting until every leaf is prime.

Worked Example 4

Write $84$84 as a product of its prime factors.

$84 = 4 \times 21 = (2 \times 2) \times (3 \times 7)$84=4×21=(2×2)×(3×7).

$84 = 2^{2} \times 3 \times 7.$84=22×3×7.

HCF and LCM by the Index (and Venn) Method

Once two numbers are in prime-factor form, you can read off the HCF and LCM directly:

• HCF: multiply the lowest power of each prime that appears in both numbers.
• LCM: multiply the highest power of each prime that appears in either number.

A Venn diagram makes this visual: shared primes go in the overlap, the rest go in the outer regions. The overlap multiplies to the HCF; the whole diagram multiplies to the LCM.

Worked Example 5

Work out the HCF and LCM of $360$360 and $84$84.

From above, $360 = 2^{3} \times 3^{2} \times 5$360=23×32×5 and $84 = 2^{2} \times 3 \times 7$84=22×3×7.

Prime	Power in $360$360	Power in $84$84	Lowest (HCF)	Highest (LCM)
$2$2	$2^{3}$23	$2^{2}$22	$2^{2}$22	$2^{3}$23
$3$3	$3^{2}$32	$3^{1}$31	$3^{1}$31	$3^{2}$32
$5$5	$5^{1}$51	—	—	$5^{1}$51
$7$7	—	$7^{1}$71	—	$7^{1}$71

$\text{HCF} = 2^{2} \times 3 = 12, \qquad \text{LCM} = 2^{3} \times 3^{2} \times 5 \times 7 = 2520.$HCF=22×3=12,LCM=23×32×5×7=2520.

Answer: HCF $= 12$=12, LCM $= 2520$=2520.

The same information as a Venn diagram:

graph LR
  subgraph "360 only"
    X["2, 5"]
  end
  subgraph "Shared (HCF)"
    Y["2, 2, 3"]
  end
  subgraph "84 only"
    Z["7"]
  end

Here the overlap $\{2, 2, 3\}${2,2,3} multiplies to $12$12 (the HCF), and all the primes together give $2^{3} \times 3^{2} \times 5 \times 7 = 2520$23×32×5×7=2520 (the LCM).

Worked Example 6

Work out the LCM of $12$12 and $18$18 by listing, then check with prime factors.

Multiples of $12$12: $12, 24, 36, 48, \ldots$12,24,36,48,… Multiples of $18$18: $18, 36, 54, \ldots$18,36,54,… The first shared multiple is $36$36.

Check: $12 = 2^{2} \times 3$12=22×3, $18 = 2 \times 3^{2}$18=2×32, so LCM $= 2^{2} \times 3^{2} = 36$=22×32=36 ✓.

The HCF × LCM Relationship

For any two positive integers $a$a and $b$b:

$\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b.$HCF(a,b)×LCM(a,b)=a×b.

This is a quick way to find one of the four quantities if you know the other three.

Worked Example 7

Two numbers have a product of $432$432 and an HCF of $6$6. Work out their LCM.

$\text{LCM} = \dfrac{a \times b}{\text{HCF}} = \dfrac{432}{6} = 72.$LCM=HCFa×b​=6432​=72.

Answer: LCM $= 72$=72.

Extended Worked Examples

Worked Example 8: A worded LCM problem

Two lighthouses flash at regular intervals. One flashes every $24$24 seconds, the other every $36$36 seconds. They flash together at midnight. Work out the next time they flash together.

This is the LCM of $24$24 and $36$36. In primes: $24 = 2^{3} \times 3$24=23×3, $36 = 2^{2} \times 3^{2}$36=22×32.

$\text{LCM} = 2^{3} \times 3^{2} = 72.$LCM=23×32=72.

They flash together every $72$72 seconds, so the next time is $72$72 seconds after midnight, i.e. at $00{:}01{:}12$00:01:12.

Worked Example 9: A worded HCF problem

A florist has $54$54 roses and $36$36 tulips. She wants to make identical bunches using all the flowers, with no flowers left over. Work out the greatest number of bunches she can make, and the contents of each.

The greatest number of equal groups is the HCF of $54$54 and $36$36. In primes: $54 = 2 \times 3^{3}$54=2×33, $36 = 2^{2} \times 3^{2}$36=22×32.

$\text{HCF} = 2 \times 3^{2} = 18.$HCF=2×32=18.

So she can make $18$18 bunches. Each bunch has $54 \div 18 = 3$54÷18=3 roses and $36 \div 18 = 2$36÷18=2 tulips.

Common error: Choosing the LCM here. "Greatest number of equal groups sharing items out" is an HCF; "next time two cycles coincide" is an LCM.

Worked Example 10: Three-number HCF and LCM

Work out the HCF and LCM of $24$24, $40$40 and $60$60.

Prime factor each: $24 = 2^{3} \times 3$24=23×3, $40 = 2^{3} \times 5$40=23×5, $60 = 2^{2} \times 3 \times 5$60=22×3×5.

HCF (lowest power present in all three): only $2$2 appears in all three, at lowest power $2^{2}$22, so HCF $= 4$=4.

LCM (highest power of each prime across all three): $2^{3} \times 3 \times 5 = 120$23×3×5=120.

Answer: HCF $= 4$=4, LCM $= 120$=120.

Answering at different grade levels

Specimen question modelled on the OCR J560 paper format: Write $600$600 as a product of its prime factors, giving your answer in index form.