# Prime and composite numbers

Here you will learn about prime and composite numbers, including what prime and composite numbers are and how to determine whether a number is prime or composite.

Students will first learn about prime and composite numbers as a part of operations and algebraic thinking in 4 th grade.

## What are prime and composite numbers?

Prime numbers are natural numbers (whole numbers) that are greater than 1 and cannot be divided by any other whole number other than itself and 1.

The number 1 is not a prime number since it only has one factor, 1.

For example, the number 7 is prime because it has 2 divisors, 1 and itself, 7.

Composite numbers are positive integers that have more than two factors.

For example, the number 12 is composite because it has 6 divisors, 1, 2, 3, 4, 6, and 12.

There are two types of composite numbers, even composite numbers and odd composite numbers. To determine whether a number is composite, you can use the following properties of composite numbers:

• A composite number has more than two factors.
• The smallest composite number is 4.
• All even numbers are composite, except for the number 2.
• The number has at least two prime factors.

### Identifying prime and composite numbers

To determine if a number is a prime number or composite number, you can use the divisibility test to find the factors of a number. If the given number has only 2 factors (1 and itself ) the number is prime. If the given number has more than 2 factors, it’s composite.

To perform a divisibility test, you can use the following divisibility rules:

## Common Core State Standards

How does this relate to 4 th 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 identify prime and composite numbers

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

1. Use the divisibility rules to see whether \bf{2, 3} or \bf{5} is a factor.
2. If they are not factors, test for divisibility by \bf{7} and \bf{8}.
3. State whether the number is prime or composite.

## Prime and composite numbers examples

### Example 1: determine if a given number is prime or composite

Is 17 prime or composite?

1. Use the divisibility rules 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 1+7=8, 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 \bf{7} and \bf{8}.

Start by dividing by 7\text{:}

As 17 \div 7=2 \; r 3, \, 7 is not a factor of 17.

Next divide by 8\text{:}

As 17 \div 8=2 \; r 1, \, 8 is not a factor of 17.

The number 17 is not divisible by 7 or 8, so there are no more whole numbers that you need to try.

3State whether the number is prime or composite.

17 is a prime number as it only has two factors, 1 and 17.

### Example 2: determine if a given number is prime or composite

Is 63 prime or composite?

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

If they are not factors, test for divisibility by \bf{7} and \bf{8}.

State whether the number is prime or composite.

### Example 3: determine if the given numbers are prime numbers or composite numbers.

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

If they are not factors, test for divisibility by \bf{7} and \bf{8}.

State whether the number is prime or composite.

### Example 4: determine if the given numbers are prime numbers or composite numbers.

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

If they are not factors, test for divisibility by \bf{7} and \bf{8}.

State whether the number is prime or composite.

### Example 5: list the composite numbers between 90 and 99.

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

If they are not factors, test for divisibility by \bf{7} and \bf{8}.

State whether the number is prime or composite.

### Example 6: list the prime numbers between 60 and 70.

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

If they are not factors, test for divisibility by \bf{7} and \bf{8}.

State whether the number is prime or composite.

### Teaching tips for prime and composite numbers

• Have students in your math classes find examples of composite numbers using divisibility rules to compile in a class list. This list can be printed and given as a list of composite numbers and a list of prime numbers to refer back to when needed.

• Instead of using worksheets where students have to identify whether a number is prime or composite, use manipulative and hands-on activities for students to sort numbers.

### Easy mistakes to make

• \bf{1} is not a prime or composite number
1 is not a prime or composite number. This is because it only has one factor, rather than the 2 or more factors needed to be prime or composite.

• All even numbers are composite
2 is the only even prime number, because it only has two factors, 1 and itself. Every other even number is divisible by 2. \, 2 is a special case when discussing prime numbers.

• All prime numbers are odd
All but one of the prime numbers are odd, it can be assumed that all odd numbers are prime. However, not all odd numbers are prime, and not all prime numbers are odd. For example, the number 15 is an odd number. The number 15 is a multiple of 3 and 5 and is composite.

### Practice prime and composite numbers questions

1) Is 49 prime or composite?

Neither

Not enough information is given

Prime

Composite

Use divisibility rules to determine whether 49 is prime or composite.

49 is not a multiple of 2, 3 or 5, so test divisibility for 7 or 8.

49 \div 7=7, so 7 is a factor of 49.

49 is composite.

2) Is 73 prime or composite?

Prime

Composite

Neither

Not enough information is given

Use divisibility rules to determine whether 73 is prime or composite.

73 is not a multiple of 2, 3 or 5, so test divisibility for 7 or 8.

73 \div 7=10 \; r3, so 7 is not a factor of 73.

73 \div 8=9 \; r1, so 8 is not a factor of 73.

73 is prime.

3) Which of the following numbers is prime?

54

82

87

97

Use divisibility rules to determine which number is prime. Prime numbers will only have 2 factors, 1 and itself.

54 and 82 are not prime because the last digits of the numbers are even, meaning 2 is a factor.

87 is not prime because when you add the digits, 8+7=15, \, 15 is divisible by 3, making 3 a factor of 87.

97 is prime.

4) Which of the following numbers is composite?

23

33

59

83

Use divisibility rules to determine which number is composite. Composite numbers have more than 2 factors.

33 is not prime, because when you add the digits, 3+3=6, 6 is divisible by 3, making 3 a factor of 33.

23, 59, and 83 only have 2 factors, 1 and itself.

33 is composite.

5) Which of the following answer choices lists all composite numbers between 20 and 30?

21, 22, 24, 25, 26, 27, 28

21, 22, 23, 25, 26, 27, 28

21, 22, 24, 25, 26, 27, 29

21, 22, 24, 26, 27, 28

Determine which of the following numbers are composite:

21, 22, 23, 24, 25, 26, 27, 28, 29.

22, 24, 26 and 28 end with even numbers, meaning that 2 is a factor.

21 has 3 as a factor because when you add the digits, 2+1=3, \, 3 is divisible by 3.

25 ends in a 5 or 0, so 5 is a factor of 25.

27 has 3 as a factor because when you add the digits, 2+7=9, \, 9 is divisible by 3.

Divide 23 and 29 by 7 and 8, using long division, until you reach the first number that gives a quotient that is a whole number.

The numbers 23 and 29 are not divisible by 7 or 8, so there are no more whole numbers that need to be tried.

The numbers 21, 22, 24, 25, 26, 27, 28 are composite numbers.

6) Which of the following answer choices lists all prime numbers between 70 and 80?

73 and 79

72, 74, and 78

71, 73 and 79

71, 73, 75

Determine which of the following numbers are composite:

71, 72, 73, 74, 75, 76, 77, 78, 79.

72, 74, 76 and 78 end with even numbers, meaning that 2 is a factor.

75 ends in a 5 or 0, so 5 is a factor of 75.

77 has 7 as a factor because 7 \times 11=77.

Divide 71, 73 and 79 by 7 and 8, using long division, until you reach the first number that gives a quotient that is a whole number.

The numbers 71, 73 and 79 are not divisible by 7 or 8, so there are no more whole numbers that need to be tried.

The numbers 71, 73 and 79 are prime numbers.

## Prime and composite numbers FAQs

What is a primality test?

A primality test is a mathematical algorithm or procedure designed to determine whether a given number is a prime number or not. The goal of such a test is to identify prime numbers without having to factorize the number completely.

What is a highly composite number?

A highly composite number is a positive integer that has more divisors than any positive integer smaller than itself. Examples of highly composite numbers include 2, 4, 6, 12, 24, 48, 60, 120 and 180.

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