Factors, Multiples and Primes
🎯 In this topic you will
- Write a positive integer as a product of prime factors
- Use prime factors to find the highest common factor (HCF) and the lowest common multiple (LCM)
🧠 Key Words
- factor tree
- highest common factor (HCF)
- index
- integer
- lowest common multiple (LCM)
- prime factor
Show Definitions
- factor tree: A diagram used to break down a number into its prime factors step by step.
- highest common factor (HCF): The largest number that divides exactly into two or more numbers.
- index: A small number that shows how many times a number is multiplied by itself (also called an exponent).
- integer: A whole number that can be positive, negative, or zero.
- lowest common multiple (LCM): The smallest number that is a multiple of two or more numbers.
- prime factor: A factor of a number that is a prime number.
🌳 Prime Factor Trees
Any integer bigger than 1:
- is a prime number, or
- can be written as a product of prime numbers.
Example:
46 = 2 × 23 47 is prime 48 = 2 × 2 × 2 × 2 × 3
49 = 7 × 7 50 = 2 × 5 × 5
You can use a factor tree to write an integer as a product of its prime factors.
This is how to draw a factor tree for 120.
- Write 120.
- Draw branches to two numbers that have a product of 120. Do not use 1 as one of the numbers. Here we have chosen 12 and 10.
120 = 12 × 10 - Do the same with 12 and 10. Here 12 = 3 × 4 and 10 = 2 × 5
- 3, 2 and 5 are prime numbers, so circle them.
- Draw two more branches from 4. 4 = 2 × 2. Circle the 2s.
- Now all the end numbers are prime, so stop.
- 120 is the product of all the end numbers: 120 = 2 × 2 × 2 × 3 × 5
- You can check that this is correct using a calculator.
You can also write the result like this: 120 = 23 × 3 × 5
23 means 2 × 2 × 2 and the small 3 is an index.
Now check that 75 = 3 × 52
You can use products of prime factors to find the HCF and LCM of two numbers.
🧠 Think like a Mathematician
Question: Can a number like 120 have more than one factor tree? Do they all lead to the same prime factors?
Equipment: Pencil, paper
- The factor tree for 120 in Section 1.1 started with $12 \times 10$.
- a) Draw a factor tree for 120 that starts with $6 \times 20$.
- b) Compare your answer to part a with a partner’s. Are your trees the same or different?
- c) Draw some different factor trees for 120. Can you say how many different trees are possible?
- d) Do all factor trees for 120 have the same end points?
Follow-up Questions:
👀 show answer
- 1: The prime factors of 120 are $2 \times 2 \times 2 \times 3 \times 5$ or $2^3 \times 3 \times 5$.
- 2: The order of factoring doesn’t matter — different trees can be drawn, but the final prime factor list is always the same.
- 3: All factor trees must end with prime factors, and every number has a unique prime factorisation (Fundamental Theorem of Arithmetic).
❓ EXERCISES
1. a) Complete this factor tree for 108.
b) Draw a different factor tree for 108.
c) Write 108 as a product of its prime factors.
d) Compare your factor trees and your product of prime factors with a partner’s.
Have you drawn the same trees or different ones? Are your trees correct?
👀 Show answer
108 → 2 × 54 → 2 × (2 × 27) → 2 × 2 × (3 × 9) → 2 × 2 × 3 × 3 × 3
So, prime factorisation: $2^2 \times 3^3$
2. a) Draw a factor tree for 200 that starts with $10 \times 20$.
b) Write 200 as a product of prime numbers.
c) Compare your factor tree with a partner’s. Have you drawn the same tree or different ones? Are your trees correct?
d) How many different factor trees can you draw for 200 that start with $10 \times 20$?
👀 Show answer
b) $2^3 \times 5^2$
c) Trees may differ but should give the same prime factorisation.
d) Several, including 10 × 20, 25 × 8, 4 × 50, 2 × 100, etc.
3. a) Draw a factor tree for 330.
b) Write 330 as a product of prime numbers.
👀 Show answer
b) $2 \times 3 \times 5 \times 11$
4. Match each number to a product of prime factors.
The first one has been done for you: a and i.
| Letter | Number | Match | Prime Factorisation |
|---|---|---|---|
| a | 20 | i | $2^2 \times 5$ |
| b | 24 | v | $2^3 \times 3$ |
| c | 42 | ii | $2 \times 3 \times 7$ |
| d | 50 | iv | $2 \times 5^2$ |
| e | 180 | iii | $2^2 \times 3^2 \times 5$ |
👀 Show answer
b → v: $2^3 \times 3$
c → ii: $2 \times 3 \times 7$
d → iv: $2 \times 5^2$
e → iii: $2^2 \times 3^2 \times 5$
5. Work out the product of each set of prime factors:
🔎 Reasoning Tip
Using factor trees: You can use a factor tree to help break down numbers into their prime factors.
b) $2^3 \times 5^3$
c) $2^2 \times 3^2 \times 11$
d) $2^4 \times 7^2$
e) $3 \times 17^2$
👀 Show answer
b) 1000
c) 396
d) 784
e) 867
6. Write each of these numbers as a product of prime factors:
a) 28
b) 60
c) 72
d) 153
e) 190
f) 275
👀 Show answer
b) $2^2 \times 3 \times 5$
c) $2^3 \times 3^2$
d) $3^2 \times 17$
e) $2 \times 5 \times 19$
f) $5^2 \times 11$
7. a) Copy the table and write each number as a product of prime numbers.
| Number | Product of prime numbers |
|---|---|
| 35 | $5 \times 7$ |
| 70 | |
| 140 | |
| 280 |
b) Add more rows to the table to continue the pattern.
👀 Show answer
| Number | Product of prime numbers |
|---|---|
| 35 | $5 \times 7$ |
| 70 | $2 \times 5 \times 7$ |
| 140 | $2^2 \times 5 \times 7$ |
| 280 | $2^3 \times 5 \times 7$ |
Pattern: powers of 2 increase by 1 each time
8. a) Write 1001 as a product of prime numbers.
b) Write 4004 as a product of prime numbers.
c) Write 6006 as a product of prime numbers.
👀 Show answer
b) 4004 = $2^2 \times 7 \times 11 \times 13$
c) 6006 = $2 \times 3 \times 7 \times 11 \times 13$
9. a) Use a factor tree to write 132 as a product of prime numbers.
b) Write 150 as a product of prime numbers.
c) $132 \times 150 = 19800$. Use this fact to write 19800 as a product of prime numbers.
👀 Show answer
b) 150 = $2 \times 3 \times 5^2$
c) 19800 = $2^3 \times 3^2 \times 5^2 \times 11$
10. a) Write each of these numbers as a product of prime numbers:
i) 15 ii) 15² iii) 28
iv) 28² v) 36 vi) 36²
b) What do you notice about your answers to i and ii, iii and iv, v and vi?
c) If $96 = 2^5 \times 3$, show how to find the prime factors of $96^2$.
Will your method work for all numbers?
👀 Show answer
ii) $(3 \times 5)^2 = 3^2 \times 5^2$
iii) $2^2 \times 7$
iv) $(2^2 \times 7)^2 = 2^4 \times 7^2$
v) $2^2 \times 3^2$
vi) $(2^2 \times 3^2)^2 = 2^4 \times 3^4$
b) When you square a number, you double the powers of each prime factor.
c) $96 = 2^5 \times 3 \Rightarrow 96^2 = 2^{10} \times 3^2$. Yes, the method works for all numbers: square each prime’s power.
11. $40 = 2 \times 2 \times 2 \times 5$ and $28 = 2 \times 2 \times 7$
Use these facts to find:
a) the HCF of 40 and 28
b) the LCM of 40 and 28
👀 Show answer
b) LCM: $2^3 \times 5 \times 7 = 280$ (all prime factors at highest powers)
12. $450 = 2 \times 3 \times 3 \times 5 \times 5$ and $60 = 2 \times 2 \times 3 \times 5$
Use these facts to find:
a) the HCF of 450 and 60
b) the LCM of 450 and 60
👀 Show answer
b) LCM = $2^2 \times 3^2 \times 5^2 = 900$
13. $180 = 2^2 \times 3^2 \times 5$ and $54 = 2 \times 3^3$
Use these facts to find:
a) the HCF of 180 and 54
b) the LCM of 180 and 54
👀 Show answer
b) LCM = $2^2 \times 3^3 \times 5 = 540$
14. a) Write 45 as a product of prime numbers.
b) Write 75 as a product of prime numbers.
c) Find the LCM of 45 and 75.
d) Find the HCF of 45 and 75.
👀 Show answer
b) 75 = $3 \times 5^2$
c) LCM = $3^2 \times 5^2 = 225$
d) HCF = $3 \times 5 = 15$
15. a) Draw factor trees to find the LCM of 90 and 140.
b) Compare your answer with a partner’s. Did you draw the same factor trees? Have you both got the same answer?
👀 Show answer
140 = $2^2 \times 5 \times 7$
LCM = $2^2 \times 3^2 \times 5 \times 7 = 1260$
Your trees may look different, but prime factorisation should be the same.
16. a) Write 396 as a product of prime numbers.
b) Write 168 as a product of prime numbers.
c) Find the HCF of 396 and 168.
d) Find the LCM of 396 and 168.
👀 Show answer
168 = $2^3 \times 3 \times 7$
HCF = $2^2 \times 3 = 12$
LCM = $2^3 \times 3^2 \times 7 \times 11 = 5544$
17. a) Find the HCF of 34 and 58.
b) Find the LCM of 34 and 58.
👀 Show answer
58 = $2 \times 29$
HCF = 2
LCM = $2 \times 17 \times 29 = 986$
18. Show that the HCF of 63 and 110 is 1.
👀 Show answer
110 = $2 \times 5 \times 11$
No common prime factors.
✅ HCF = 1
19. 37 and 47 are prime numbers.
a) What is the HCF of 37 and 47?
b) What is the LCM of 37 and 47?
c) Write a rule for finding the HCF and LCM of two prime numbers.
d) Compare your answer to part c with a partner’s answer.
Check your rules by finding the HCF and LCM of 39 and 83.
👀 Show answer
b) LCM = $37 \times 47 = 1739$
c) For two prime numbers:
– HCF is always 1
– LCM is the product of the two numbers
d) 39 = $3 \times 13$, 83 is prime → no common factor
HCF = 1, LCM = $39 \times 83 = 3237$
⚠️ Be careful!
Don’t mix up the rules for finding the HCF and the LCM using prime factors. For the HCF, use the lowest powers of the common primes. For the LCM, use the highest powers of all primes found in either number.