# Prime Numbers

Here we will learn about prime numbers, including what they are, how we can determine whether a number is prime, and how to solve problems that involve prime numbers.

There are also prime number worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

## What are prime numbers?

Prime numbers are positive integers (whole numbers) that have only two factors, themselves and 1.

This means that you cannot divide a prime number by any number apart from 1 or itself, and get an integer answer.

A number that is not prime is called a composite number.

There are 25 prime numbers between 1 and 100. These are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29,  31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.

Notice that,

1 is not a prime number as it has only 1 factor.

2 is the only even prime number.

You should be able to recall the first 8 prime numbers (up to 20 ) and be able to determine whether a number is prime by looking at the factors of the number.

To determine whether the number is prime, check whether it has any factors other than itself and 1, either manually or by using a number trick. If the number has a factor that is not itself or 1, it is not prime.

There are a few useful number tricks that can help us determine whether a number is divisible by an integer and therefore has that integer as a factor:

Once we have considered these number tricks, we need to look for divisibility by other numbers. We don’t, however, need to consider every number.

The reason for this is that for a number to be divisible by 4 or 6 or any multiple of 2, it must also be divisible by 2, which is something we have already checked.

For a number to be divisible by 6 or 9 or any multiple of 3, it must also be divisible by 3, which is something that we have already checked.

This pattern continues and means that we only need to check for divisibility by other prime numbers, starting with the number 7.

## How to find prime numbers

In order to determine whether a number is a prime number:

1. Use the number tricks to see whether \bf{2, 3} or \bf{5} is a factor.
2. If they are not factors, test for divisibility by other prime numbers, starting with \bf{7}.
3. State your conclusion with a reason.

### Related lessons onfactors, multiples and primes

Prime numbers is part of our series of lessons to support revision on factors, multiples and primes. You may find it helpful to start with the main factors, multiples and primes lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

## Prime numbers examples

### Example 1: two digit number

Is 53 a prime number?

1. Use the number tricks to see whether \bf{2, 3} or \bf{5} is a factor.

The last digit is not a 2, 4, 6, 8, or 0 so it is not a multiple of 2.

Adding the digits together we have 5+3=8 and so it is not a multiple of 3.

The last digit is not a 5 or a 0 and so it is not a multiple of 5.

2If they are not factors, test for divisibility by other prime numbers, starting with \bf{7}.

We need to continue dividing 53 by successive prime numbers until we reach the first prime number that is greater than \sqrt{53}.

\sqrt{53} \approx 7.3 so we only need to try prime numbers less than this.

Start by dividing by 7:

As 53\div{7}=7.57142857 \ 7 is not a factor of 53.

The next prime number is 11, which is greater than 7.3 so there are no more integers that we need to try.

3State your conclusion with a reason.

53 is a prime number as it only has two factors, 1 and 53.

### Example 2: three digit number

Is 223 a prime number?

Use the number tricks to see whether \bf{2, 3} or \bf{5} is a factor.

If they are not factors, test for divisibility by other prime numbers, starting with \bf{7}.

State your conclusion with a reason.

### Example 3: four digit number

Is 2073 a prime number?

Use the number tricks to see whether \bf{2, 3} or \bf{5} is a factor.

If they are not factors, test for divisibility by other prime numbers, starting with \bf{7}.

State your conclusion with a reason.

### Example 4: determine the prime number from a list of numbers

One of the following numbers is prime. Identify the prime number.

Use the number tricks to see whether \bf{2, 3} or \bf{5} is a factor.

If they are not factors, test for divisibility by other prime numbers, starting with \bf{7}.

State your conclusion with a reason.

## Problem solving with prime numbers

In order to place numbers into a diagram / table:

1. Determine what each box / section of the diagram represents.
2. Place each number one-by-one into the diagram.
3. Determine the frequency of values in each box / section if required.

## Problem solving with prime numbers examples

### Example 5: complete the diagram (two way table)

Place the set of numbers N=\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} into the two-way table below.

Determine what each box / section of the diagram represents.

Place each number one-by-one into the diagram.

Determine the frequency of values in each box / section if required.

### Example 6: complete the diagram (Venn diagram)

The Venn diagram below represents the two sets P=\{ Prime \} and F=\{ Factor of 24 \}.

Given that ξ=\{x is an integer such that 0 < x \leq 24 \}, complete the Venn diagram.

Use your Venn diagram to write down the frequency of values in the sets.

i) P

ii) P \cap F

Determine what each box / section of the diagram represents.

Place each number one-by-one into the diagram.

Determine the frequency of values in each box / section if required.

### Common misconceptions

• \bf{1} is not a prime number

1 is not a prime number. This is because it only has one factor, rather than the 2 factors needed to be a prime number.

• \bf{2} is a prime number

2 is the only even prime number. This is because it only has two factors, 1 and itself, and therefore by definition, it is a prime number. Every other even number is divisible by 2. \ 2 is therefore known as a special case when discussing prime numbers.

• Prime numbers and the odd numbers

As all but one of the prime numbers are odd (remember that 2 is the only even prime number), it is sometimes assumed that all odd numbers are prime. Take the number 9. It is an odd number, however 9 is a multiple of 3 and so we can divide 9 by 3 (and get 3 ). Not all odd numbers are prime, and not all prime numbers are odd.

• Decimals cannot be prime numbers

All prime numbers are positive integers but a common misconception is that decimals can be prime. For example, the decimal 2.3 is considered to be a prime number as 23 is a prime number.

### Practice prime numbers questions

1. Determine what type of number 71 is from the list below.

Multiple of 3

Multiple of 4

Multiple of 5

Prime number

71 does not end in 0, 2, 4, 6 or 8 and so is not a multiple of 2. It cannot therefore be a multiple of 4.

7+1=4 so 71 is not a multiple of 3 .

71 does not end in 0 or 5 therefore it is not a multiple of 5.

71 \div 7=10.14 therefore it is not a multiple of 7.

This tells us that 71 is a prime number.

2. Determine what type of number 513 is from the list below.

Multiple of 2

Multiple of 3

Multiple of 5

Prime number

513 is a multiple of 3 as 5+1+3=9 which is a multiple of 3.

513\div{3}=171

3. Determine what type of number 143 is from the list below.

Multiple of 3

Multiple of 7

Multiple of 11

Prime

1+4+3=8 so 143 is not a multiple of 3 .

143\div 7=20.42857… so 143 is not a multiple of 7 .

143\div{11}=13 so 143 is a multiple of 11 .

4. Which number from the list is prime?

81

343

2

3\times{10}^{2}

2 is the only even prime number. It has two factors, 1 and itself.

81 has factors of 1, 3, 9, 27 and 81.

343 has factors of 1, 7, 49 and 343.

3\times{10}^{2} written as an ordinary number is 300 which has many factors.

5. Let N=\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}. Which two way table correctly sorts the set N?

Factors of 12: 1, 2, 3, 4, 6, 12

Prime Numbers: 2, 3, 5, 7, 11

6. The Venn diagram below represents the two sets P=\{ Prime \} and F=\{ Factor of 10\}. Given that ξ= \{x is an integer such that 0 < x \leq 12 \}, work out the frequency of values that would appear in P \cap F.

1

2

2 and 5

10

Factors of 10: 1, 2, 5, and 10.

Prime Numbers: 2, 3, 5, and 7.

Each value placed correctly in the Venn diagram gives the result:

The frequency that corresponds to each set is therefore:

### Prime numbers GCSE questions

1. (a) Write the first 8 prime numbers.

(b) Peter says “every prime number is odd”. Is Peter correct? Explain your answer.

(3 marks)

(a)

2, 3, 5, 7, 11, 13, 17, 19

(1)

(b)

No

(1)

2 is the only even prime number.

(1)

2. (a) The perimeter of a rectangle can be written as P=2(l+w) where l and w are integer side lengths.

Given that the area of the rectangle is 23cm^{2}, calculate the perimeter of the rectangle.

(b) A square has an area of A=x^{2}, where x is an integer. Is the area always, sometimes or never a prime number? Explain your answer.

(4 marks)

(a)

l=1 and w=23 or l=23 and w=1

(1)

P=2(23+1)=2\times{24}=48\text{cm}

(1)

(b)

All square numbers have an odd number of factors, whereas all prime numbers have 2 factors.

or

1 is not prime and any other square number has a factor of its square root meaning it is not prime.

(1)

Never

(1)

3.  Below is a list of numbers:

(a) Which of the values is a prime number?

(b) Which of the values is 2 more than a prime number, and 2 less than another prime number?

(c) Which of the values gives a prime number when it is squared?

(3 marks)

(a) 73

(1)

(b) 21

(1)

(c) \sqrt{5}

(1)

4. (a) On the Venn diagram below, P=\{ Prime numbers \} and M=\{ Multiples of 4\}.

For the set of numbers ξ=\{x such that 20 < x \leq 40 \}, complete the Venn diagram.

(b) Write down the numbers that are in the set P .

(c) How many numbers are in the set P \cap M?

(6 marks)

(a)

Prime Numbers: 23, 29, 31, 37.

(1)

Multiples of 4: 20, 24, 28, 32, 36, 40.

(1)

No values in the intersection (P \cap M).

(1)

(P \cup M)’: 21, 22, 25, 26, 27, 30, 33, 34, 35, 38, 39.

(1)

(b) 23, 29, 31, 37

(1)

(c) 0

(1)

## Learning checklist

You have now learned how to:

• Identify prime numbers
• Know and use the vocabulary of prime numbers, prime factors and composite (non-prime) numbers
• Establish whether a number up to 100 is prime and recall prime numbers up to 19

## The next lessons are

The sieve of Eratosthenes