# 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.

Students will first learn about prime numbers as part of operations and algebraic thinking in elementary school.

## What are prime numbers?

Prime numbers are 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 a whole number as an answer.

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

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

2 is the only even prime 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 divisibility rule. If the number has a factor that is not itself or 1, it is not prime.

There are a few useful divisibility rules that can help us determine whether a number is divisible by a whole number and therefore has that whole number as a factor.

## Common Core State Standards

How does this relate to 4th grade math and 5th grade math?

• Grade 4: Operations and Algebraic Thinking (4.OA.B.4)
Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1– 100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite.

## How to find prime numbers

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

1. Use the divisibility rules to see whether 2, 3 or 5 is a factor.
2. If they are not factors, test for divisibility by 7 and 8.
3. State your conclusion with a reason.

## Prime numbers examples

### Example 1: two digit number

Is 53 a prime number?

1. Use the divisibility rules to see whether 2, 3 or 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 7 and 8.

We need to divide 53 by 7 and 8, using long division, until we reach the first number that gives a quotient that is a whole number.

Start by dividing by 7\text{:}

As 53 \div 7=7 \; r4 \; 7 is not a factor of 53.

Next divide by 8\text{:}

As 53 \div 8=6 \; r5 \; 8 is not a factor of 53

The number 53 is not divisible by 7 or 8, so there are no more whole numbers 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: determine the prime number from a list of numbers

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

Use the divisibility rules to see whether 2, 3 or 5 is a factor.

If they are not factors, test for divisibility by 7 and 8.

State your conclusion with a reason.

### Teaching tips for prime numbers

• Fluency with multiplication facts can assist students when trying to determine if a number is a composite or prime number.

• Instead of simply giving students the option to choose if a number is a prime number, consider having them explain their thinking behind their choice.

• Proving the divisibility rules at the beginning will help students become familiar with the tips to find prime numbers.

### Easy mistakes to make

• 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.

• 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 whole numbers 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 \; r1 therefore it is not a multiple of 7.

This tells us that 71 is a prime number.

2. Determine what type of number 55 is from the list below.

Multiple of 2

Multiple of 3

Multiple of 5

Prime number

55 is a multiple of 5 as the last digit is either a 5 or a 0.

55 \div 5=11

3. Determine what type of number 49 is from the list below.

Multiple of 3

Multiple of 6

Prime

Multiple of 7

4+9=13, which is not divisible by 3, so 49 is not a multiple of 3.

49 \div 8=6 \; r 1 so 49 is not a multiple of 6.

49 \div 7=7 so 49 is a multiple of 7.

4. Which number from the list is prime?

81

78

2

3 \times 10

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

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

78 has factors of 1, 7, 49 and 343

3 \times 10 written as an ordinary number is 30 which has many factors.

5. Which number from the list is not prime?

37

76

83

29

37, 83, and 29 only have two factors, 1 and itself. Therefore these numbers are prime.

76 has the factors of 1, 2, 4, 19, 38, and 76. This number is composite, not prime.

6. Which number from the list is prime?

39

21

93

57

39 has the factors of 1, 3, 13, 39.

21 has the factors of 1, 3, 7, 21.

93 has the factors of 1, 3, 31, 93.

57 only has 2 factors, 1 and itself.

57 is prime.

## Prime numbers FAQs

Are all prime numbers odd?

With the exception of 1 and 2, all odd numbers are prime. Because all even numbers are divisible by 2, they are composite. 1 is neither prime nor composite.

Why is 1 not considered prime?

1 is not prime or composite because it only has one factor: 1.

## The next lessons are

The sieve of eratosthenes

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