šŸ”¢ Integers: Factors, Multiples & Primes

Cambridge Lower Secondary Ā· Grade 8 Ā· Number & Calculation

Prime Factorisation
360 = 2Ā³ Ɨ 3Ā² Ɨ 5
Every integer has a unique prime factorisation
HCF & LCM from Venn
HCF = product of the overlap
LCM = product of everything
Venn diagrams make it visual
Real-World LCM
Bus A every 12 min, Bus B every 18 min
Next together: LCM(12,18) = 36 min

Watch a factor tree grow! šŸŒ³

360 8 45 2 4 9 5 360 = 2Ā³ Ɨ 3Ā² Ɨ 5

What you'll learn:

  • Prime factorisation ā€” factor trees & ladder method
  • Index form: writing factorisations as 2Ā³ Ɨ 3Ā² Ɨ 5
  • HCF using prime factorisation and Venn diagrams
  • LCM using prime factorisation and Venn diagrams
  • HCF and LCM of three numbers
  • Checking if large numbers are prime (trial division)
  • Applying HCF/LCM: tiles, buses, gears and more
Created by Mr Josh

šŸ“– Learn: Factors, Multiples & Primes

Part 1: Prime Factorisation

Every integer greater than 1 can be written as a product of prime numbers. This is called its prime factorisation and it is unique (Fundamental Theorem of Arithmetic).

Method 1 ā€” Factor Tree: Split the number into any two factors, then keep splitting until every branch ends in a prime (circle it).

Example: 120 ā†’ 12 Ɨ 10 ā†’ (4 Ɨ 3) Ɨ (2 Ɨ 5) ā†’ (2 Ɨ 2 Ɨ 3) Ɨ (2 Ɨ 5) = 2Ā³ Ɨ 3 Ɨ 5

Method 2 ā€” Ladder (Repeated Division): Divide by the smallest prime possible at each step.

120 Ć· 2 = 60
60 Ć· 2 = 30
30 Ć· 2 = 15
15 Ć· 3 = 5
5 Ć· 5 = 1  ā†’  2Ā³ Ɨ 3 Ɨ 5
šŸ’” Always start dividing by 2, then 3, then 5, 7, 11ā€¦ in order. Write primes on the left, results on the right. Stop when you reach 1.

Part 2: Index Form

When the same prime appears multiple times, use powers (index form):

2 Ɨ 2 Ɨ 2 Ɨ 3 Ɨ 3 Ɨ 5 = 2Ā³ Ɨ 3Ā² Ɨ 5
Example: 1764 = 2 Ɨ 2 Ɨ 3 Ɨ 3 Ɨ 7 Ɨ 7 = 2Ā² Ɨ 3Ā² Ɨ 7Ā²
Example: 2310 = 2 Ɨ 3 Ɨ 5 Ɨ 7 Ɨ 11 = 2 Ɨ 3 Ɨ 5 Ɨ 7 Ɨ 11 (all different)
NumberFactor TreeIndex Form
722Ɨ2Ɨ2Ɨ3Ɨ32Ā³ Ɨ 3Ā²
3602Ɨ2Ɨ2Ɨ3Ɨ3Ɨ52Ā³ Ɨ 3Ā² Ɨ 5
25202Ɨ2Ɨ2Ɨ3Ɨ3Ɨ5Ɨ72Ā³ Ɨ 3Ā² Ɨ 5 Ɨ 7

Part 3: HCF using Venn Diagrams

The Highest Common Factor (HCF) is the largest number that divides exactly into both (or all) given numbers.

Venn Diagram Method:

  1. Write the prime factorisation of each number.
  2. Draw two overlapping circles, one for each number.
  3. Place shared prime factors in the overlap.
  4. Place remaining prime factors in the outer sections.
  5. HCF = product of the factors in the overlap.
Find HCF(72, 120):
72 = 2Ā³ Ɨ 3Ā²  |  120 = 2Ā³ Ɨ 3 Ɨ 5
Overlap: 2Ā³ Ɨ 3  ā†’  HCF = 24
šŸ’” Take the LOWER power of each shared prime factor for the HCF.

Part 4: LCM using Venn Diagrams

The Lowest Common Multiple (LCM) is the smallest number that is a multiple of both (or all) given numbers.

Using the same Venn diagram: LCM = product of ALL factors in both circles (each prime factor counted once at its highest power).

Find LCM(72, 120):
72 = 2Ā³ Ɨ 3Ā²  |  120 = 2Ā³ Ɨ 3 Ɨ 5
All factors: 2Ā³ Ɨ 3Ā² Ɨ 5  ā†’  LCM = 360
šŸ’” Take the HIGHER power of each prime factor for the LCM.

Quick check: HCF Ɨ LCM = product of the two original numbers.
24 Ɨ 360 = 8640   and   72 Ɨ 120 = 8640 āœ“

Part 5: HCF & LCM of Three Numbers

For three numbers, use a triple Venn diagram or just work from the prime factorisations directly:

Find HCF(36, 60, 84):
36 = 2Ā² Ɨ 3Ā²  |  60 = 2Ā² Ɨ 3 Ɨ 5  |  84 = 2Ā² Ɨ 3 Ɨ 7
Common to all three: 2Ā² Ɨ 3  ā†’  HCF = 12
Find LCM(36, 60, 84):
Highest powers: 2Ā² Ɨ 3Ā² Ɨ 5 Ɨ 7  ā†’  LCM = 1260
šŸ’” HCF: only primes present in ALL numbers (lowest power). LCM: primes present in ANY number (highest power).

Part 6: Is a Large Number Prime?

To check if n is prime, test divisibility by every prime up to āˆšn. If none divide exactly, it is prime.

Is 137 prime?   āˆš137 ā‰ˆ 11.7, so test 2, 3, 5, 7, 11
137 Ć· 2 = 68.5 āœ—   137 Ć· 3 = 45.67 āœ—   137 Ć· 5 = 27.4 āœ—   137 Ć· 7 = 19.57 āœ—   137 Ć· 11 = 12.45 āœ—
137 is PRIME
Is 221 prime?   āˆš221 ā‰ˆ 14.9, so test 2, 3, 5, 7, 11, 13
221 Ć· 13 = 17 āœ“  ā†’  221 = 13 Ɨ 17, NOT prime
šŸ’” You only need to test primes up to āˆšn ā€” any factor larger than āˆšn would pair with one smaller than āˆšn.
Created by Mr Josh

šŸ’” Worked Examples

Example 1: Prime Factorisation in Index Form

Express 1080 as a product of prime factors in index form.

Ladder method:
1080 Ć· 2 = 540
540 Ć· 2 = 270
270 Ć· 2 = 135
135 Ć· 3 = 45
45 Ć· 3 = 15
15 Ć· 3 = 5
5 Ć· 5 = 1
1080 = 2Ā³ Ɨ 3Ā³ Ɨ 5
Check: 8 Ɨ 27 Ɨ 5 = 8 Ɨ 135 = 1080 āœ“

Example 2: HCF using Prime Factorisation

Find the HCF of 252 and 360.

252 = 2Ā² Ɨ 3Ā² Ɨ 7
360 = 2Ā³ Ɨ 3Ā² Ɨ 5
Shared primes (lowest powers): 2Ā² and 3Ā²
HCF = 2Ā² Ɨ 3Ā² = 4 Ɨ 9 = 36
šŸ’” HCF(252, 360) = 36 means 36 is the largest tile that fits both a 252 cm and 360 cm wall exactly.

Example 3: LCM using Prime Factorisation

Find the LCM of 252 and 360.

252 = 2Ā² Ɨ 3Ā² Ɨ 7  |  360 = 2Ā³ Ɨ 3Ā² Ɨ 5
Take highest power of each prime: 2Ā³, 3Ā², 5, 7
LCM = 2Ā³ Ɨ 3Ā² Ɨ 5 Ɨ 7 = 8 Ɨ 9 Ɨ 5 Ɨ 7 = 2520
Verify: 252 Ɨ 360 = 90720 = 36 Ɨ 2520 āœ“ (HCF Ɨ LCM = product of numbers)

Example 4: Word Problem ā€” Gear Rotations

Gear A has 48 teeth and Gear B has 36 teeth. They start with a marked tooth aligned. How many full rotations does each gear make before the marks align again?

48 = 2ā“ Ɨ 3  |  36 = 2Ā² Ɨ 3Ā²
LCM = 2ā“ Ɨ 3Ā² = 16 Ɨ 9 = 144 teeth
Gear A rotations: 144 Ć· 48 = 3 rotations
Gear B rotations: 144 Ć· 36 = 4 rotations
The marks align again after Gear A makes 3 full rotations and Gear B makes 4 full rotations.
šŸ”§ LCM gives the first time two cyclic events coincide ā€” perfect for gears, buses, and rotating schedules.
Created by Mr Josh

šŸŒ³ Visualizer

Interactive Factor Tree Builder

Enter a number and click Start Tree. Then click any composite (non-prime) node to split it into two factors!

Venn Diagram: HCF & LCM

Enter two numbers, then drag each prime factor chip into the correct region of the Venn diagram.
Left only (factors of A only) Ā· Overlap (shared factors) Ā· Right only (factors of B only)

Number A:
Number B:
A B ā† Drag chips here ā†’

šŸšŒ Bus Problem Simulator

Set two bus intervals. The simulator shows when both buses next leave at the same time ā€” that's the LCM!

šŸšŒ
šŸšŽ

šŸ” Prime Number Checker

Type any number and watch trial division test each prime up to āˆšn step by step.

Created by Mr Josh

āœļø Exercise 1 ā€” Prime Factorisation

Express each number as a product of prime factors in index form.

Created by Mr Josh

āœļø Exercise 2 ā€” HCF of Two Numbers

Find the HCF of each pair of numbers using prime factorisation.

Created by Mr Josh

āœļø Exercise 3 ā€” LCM of Two Numbers

Find the LCM of each pair of numbers using prime factorisation.

Created by Mr Josh

āœļø Exercise 4 ā€” HCF & LCM of Three Numbers

For each set of three numbers, find both the HCF and the LCM.

Created by Mr Josh

šŸšŒ Exercise 5 ā€” Word Problems

Apply HCF and LCM to solve real-world problems.

Created by Mr Josh

šŸ“ Practice ā€” 20 Questions

Mixed practice across all topics. Show your working on paper!

Created by Mr Josh

šŸ† Challenge ā€” 8 Questions

Harder multi-step problems. Take your time!

Created by Mr Josh