๐ฌ See factors pair up
Factor pairs multiply to give the same whole number.
๐ฏ By the end of this lesson...
You should be able to:
- Find the factors of a whole number.
- List multiples of a number.
- Explain the difference between factors and multiples.
- Recognise prime and composite numbers.
- Explain why 1 is neither prime nor composite.
- Write a number as a product of prime factors.
- Use a factor tree correctly.
๐ค Think About This...
Two students are asked:
One student says:
The other student says:
Which student is correct?
1๏ธโฃ What Is a Factor?
A factor is a whole number that divides another whole number exactly, leaving no remainder.
Since the division gives a whole-number answer with no remainder, 3 is a factor of 12.
Since there is a remainder, 5 is not a factor of 12.
Divide the number by the possible factor.
If the answer is a whole number with no remainder, it is a factor.
๐ Finding All the Factors
To find all the factors of a number, look for pairs of whole numbers that multiply to give that number.
Example 1: Find all the factors of 24
Begin with 1 and test the multiplication pairs.
Therefore:
1, 2, 3, 4, 6, 8, 12, 24
๐ง StudyNest Method: Finding Factors
Step 1: Begin with 1, because 1 is a factor of every whole number.
Step 2: Test 2, 3, 4 and the following whole numbers in order.
Step 3: Write every exact division as a factor pair.
Step 4: Stop when the factor pairs begin repeating in reverse order.
Step 5: List all the factors from smallest to largest.
2๏ธโฃ What Is a Multiple?
A multiple is a number produced by multiplying a given number by a whole number.
For example, the multiples of 5 begin:
This is because:
๐ง StudyNest Method: Finding Multiples
Step 1: Write the number.
Step 2: Multiply it by 1, 2, 3, 4 and so on.
Step 3: Continue until you have listed the number of multiples requested.
Example 2: Write the first six positive multiples of 7
โ๏ธ Factors and Multiples: What Is the Difference?
Factors
Factors divide into a number exactly.
1, 2, 3, 4, 6, 12
A number has a limited number of factors.
Multiples
Multiples are produced by multiplying a number.
12, 24, 36, 48, โฆ
A number has infinitely many multiples.
Factors fit inside a number.
Multiples move upwards from a number.
3๏ธโฃ Prime Numbers
A prime number is a whole number greater than 1 that has exactly two positive factors:
For example, the factors of 7 are:
Since 7 has exactly two factors, 7 is prime.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, โฆ
โญ Why Is 2 Special?
The factors of 2 are:
Therefore, 2 is prime.
Every other even number is divisible by 2 and therefore has more than two factors.
4๏ธโฃ Composite Numbers
A composite number is a whole number greater than 1 that has more than two positive factors.
For example, the factors of 12 are:
Since 12 has more than two factors, it is composite.
| Number | Positive factors | Classification |
|---|---|---|
| 7 | 1, 7 | Prime |
| 12 | 1, 2, 3, 4, 6, 12 | Composite |
| 19 | 1, 19 | Prime |
| 20 | 1, 2, 4, 5, 10, 20 | Composite |
โ Is 1 Prime or Composite?
The number 1 has only one positive factor:
A prime number must have exactly two positive factors.
A composite number must have more than two positive factors.
๐ง StudyNest Method: Prime or Composite?
If not, it is neither prime nor composite.
Step 2: Find or test its factors.
Step 3: Count the factors.
Exactly two factors means the number is prime.
More than two factors means the number is composite.
5๏ธโฃ Prime Factors
A prime factor is a factor that is also a prime number.
Consider the factors of 18:
The prime numbers in this list are 2 and 3.
๐ณ Prime Factorisation
Prime factorisation means writing a composite number as a product of prime numbers only.
Example 3: Write 36 as a product of prime factors
The final branches are all prime numbers.
Using index notation:
๐ง StudyNest Method: Using a Factor Tree
Step 2: Check each factor.
If a factor is prime, circle or mark it and stop splitting it.
If it is composite, split it again.
Step 3: Continue until every final branch is prime.
Step 4: Multiply the prime factors together.
Step 5: Use powers to shorten repeated factors where appropriate.
๐ More Worked Examples
Example 4: Write 60 as a product of prime factors
Therefore:
Example 5: Is 51 prime?
Test whether 51 is divisible by smaller prime numbers.
Since 6 is divisible by 3, 51 is divisible by 3.
Example 6: Find all the factors of 36
1, 2, 3, 4, 6, 9, 12, 18 and 36.
๐ Looking Ahead: HCF and LCM
Factors and multiples help us find two important values:
Highest Common Factor
The largest factor shared by two or more numbers.
Lowest Common Multiple
The smallest positive multiple shared by two or more numbers.
โ Common Mistakes
Confusing factors with multiples.
Factors divide into a number. Multiples are produced by multiplication.
Forgetting 1 and the number itself when listing factors.
Saying that 1 is prime.
One has only one positive factor, so it is neither prime nor composite.
Thinking every odd number is prime.
For example, 9 = 3 ร 3, so 9 is composite.
Ending a factor tree before every branch is prime.
Missing a factor pair when listing all the factors of a number.
๐ฎ Your Turn!
- List all the factors of 18.
- List all the factors of 40.
- Write the first six positive multiples of 9.
- Write the first five positive multiples of 12.
- State whether 17 is prime or composite.
- State whether 27 is prime or composite.
- Explain why 1 is neither prime nor composite.
- Write 48 as a product of prime factors.
- Write 90 as a product of prime factors.
- Is 57 prime? Show how you know.
โ Show Answers
- 1, 2, 3, 6, 9, 18
- 1, 2, 4, 5, 8, 10, 20, 40
- 9, 18, 27, 36, 45, 54
- 12, 24, 36, 48, 60
- Prime, because its only positive factors are 1 and 17.
- Composite, because 27 = 3 ร 9.
- It has only one positive factor.
- 48 = 2โด ร 3
- 90 = 2 ร 3ยฒ ร 5
- No. 57 = 3 ร 19, so it is composite.
๐ง Remember This
A multiple is produced by multiplication.
A prime number has exactly two positive factors.
A composite number has more than two positive factors.
The number 1 is neither prime nor composite.
Prime factorisation ends only when every factor is prime.
๐ฏ Before Moving On...
- โ Can I explain the difference between a factor and a multiple?
- โ Can I list all the factors of a number?
- โ Can I write the first few multiples of a number?
- โ Can I identify prime and composite numbers?
- โ Can I explain why 1 is neither prime nor composite?
- โ Can I use a factor tree?
- โ Can I write a number as a product of prime factors?
๐ Well Done!
You can now work confidently with factors, multiples, prime numbers and prime factorisation.
Divide to test factors โ Multiply to list multiples โ Count factors to classify a number
๐ Your Progress
Lesson 2 of 17