# Prime Numbers, Composite Numbers, and Prime Factorization

*Ryan Bernido is an educator and writer. He enjoys math and likes explaining it to his students and readers.*

## What Are Prime and Composite Numbers?

**Prime numbers** and **composite numbers** are types of numbers based on the number of factors that a number has. A **factor** is a value or a number that can divide a number or an expression evenly, which means that when a number is divided by its factor, no remainder exists.

Prime numbers is a set of numbers where each number has only two factors (1 and itself). Composite numbers is a set of numbers where each number has more than two factors.

## Prime Number

A prime number is a number whose factors are 1 and itself only. However, the number 1 is not considered a prime number. Examples of prime numbers are 2, 3, 5, 7, 11, 13, and 17. See the table below.

## Sieve of Eratosthenes: Finding Prime Numbers 1-100

Eratosthenes was a Greek Mathematician in the 3rd Century B.C. who introduced the Sieve of Eratosthenes, his simple method of finding prime numbers from 1 through 100. The steps are as follow:

- Place all the numbers 1 through 100 in a table.
- Cross out 1 as it is not a prime number or a composite number.
- Encircle two (2). All numbers divisible by two are even numbers. So cross out all the multiples of 2.
- Encircle three (3). Cross out all multiples of three.
- Encircle five (5). Cross out all multiples of five.
- Encircle seven (7). Cross out all multiples of seven.
- Encircle eleven (11). Cross out all multiples of eleven.
- Encircle seventeen (17). Cross out all multiples of seventeen.
- Continue the process till all the numbers in the list are either encircle or crossed out. All encircled numbers from 1-100 using the Sieve of Eratosthenes are prime numbers.

The prime numbers from 1 to 100 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.

## Composite Number

A composite number is a number that has more than two factors. One (1) is not a composite despite not being considered a prime number. A composite number can be an odd composite number or even composite number. An odd composite number is a number that is an odd number and not a prime number. An even composite number is a number that is an even number and not prime number.

Examples of composite numbers are as follow (see the table below):

## Methods in Finding Prime Numbers or Composite Numbers

There are several ways to test or determine whether a number is prime or composite. Some methods are the following:

- Use of division method and divisibility rules
- Prime factorization

Per the definition of prime numbers and composite numbers, we can identify the given number. If the factors are 1 and itself only, then the number is prime. If the number has more than two factors, then the number is composite.

Let’s try prime factorization below.

## Prime Factorization

Prime factorization is a method to find the prime factors of a given number. If the prime factorization yields in the product of two or more prime numbers, then the number is a composite number. See examples below.

*Example 1:*

Find the prime factors of 36.

As we can see, the prime factors of 36 are 2, 2, 3, and 3.

Hence, we can write 36 = 2 x 2 x 3 x 3 or 36 = 2^{2} x 3^{2} . Since, 36 has more than two factors, 36 is a composite number.

Some people can have different ways of prime factorization by using different factors in the first step. See the prime factorization of 36 below.

The given prime factorization yields similar results to the previous prime factorization of 36.

*Example 2:*

Find the prime factors of 80.

The prime factors of 80 are 2, 2, 2, 2, and 5. Hence, we can write 80 = 2^{4} x 5.

Since 80 has more than two factors, 80 is a composite number.

## Practice Task 1

**Instructions:** Determine whether the given number is a prime number or composite number.

- 269
- 239
- 47
- 272
- 568
- 293
- 149
- 131
- 601
- 428

## Practice Task 2

**Instructions:** Show the prime factorization of each following number. State its prime factorization in exponential notation. Make a conclusion whether the number is a composite number or a prime number.

- 200
- 421
- 294
- 149
- 602

**Challenge!**

Can you find prime numbers from 101 to 500?

What are the prime numbers between 101 and 500?

What are the composite numbers between 101 and 500?

What method/s did you use to find the prime and composite numbers between 101 and 500?

**References **

Admin. (2021, October 7). *Prime and composite numbers - definition, examples, list and table*. BYJUS. Retrieved April 4, 2022, from https://byjus.com/maths/prime-and-composite-numbers/#:~:text=A%20prime%20number%20is%20a,a%20prime%20or%20composite%20number.

Eratosthenes sieve to find prime numbers - CBSE class 6 math notes. (n.d.). Retrieved April 4, 2022, from https://edusaksham.com/chapters/CBSE-Class-6-Mathematics-Eratosthenes-Sieve-to-Find-Prime-Numbers-Notes.html#:~:text=Prime%20Numbers%20from%201%20to,called%20the%20Sieve%20of%20Eratosthenes.

*This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.*

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