How to Determine If a Number Is Prime

A prime number can only be divided by itself and 1.
A prime number can only be divided by itself and 1. Robert Brook/Getty Images

Prime numbers are numbers that are bigger than one and cannot be divided evenly by any other number except 1 and itself. If a number can be divided evenly by any other number not counting itself and 1, it is not prime and is referred to as a composite number.

Prime numbers are whole numbers that must be greater than one, and as a result, zero and one are not considered prime numbers, nor are any number less than zero; the number two, however, is the first prime number as it can only be divided by itself and the number one.

There are a variety of methods to finding out whether a whole number is prime or not. Using a process called factorization, mathematicians can break larger numbers into the factors that can be combined to make those numbers. If more than two results (1 and the number itself) exist, the number is not prime. Students may also use calculators or separate piles of counting objects like beans or coins to determine if a number is prime.

Using Factorization to Determine if a Number Is Prime

Using a process called factorization, mathematicians can easily determine whether or not numbers are prime, but first one must understand what a factor of a number is. A factor is any number that can be multiplied by another number to get the same result.

For instance, the prime factors of the number 10 are 2 and 5 because these whole numbers can be multiplied by one another to equal 10. However, 1 and 10 are also considered factors of 10 because they can be multiplied by one another to equal 10, though this is expressed in the prime factors of 10 as 5 and 2 since both 1 and 10 are not prime numbers.

This can also be illustrated through an easier method of working with numbers in a concrete sense by giving students counting devices like beans, buttons, or coins and starting by counting out a number of those objects less than 100 then attempting to divide these new piles into equal and smaller piles of each of the prime number one to 10.

Using a Calculator and Divisibility to Determine if a Number Is Prime

After using the concrete method (buttons, coins etc.) and trying to separate the 17 or 23 coins evenly into 2 or 3 piles, then try the calculator method. After all, with any concept, concrete methods should be used before automated methods!

Take your calculator and key in the number you are trying to determine is prime by first dividing the number by two then by three to see if the result is a rounded whole number. Let's take 57 and first divide it by 2. Does it come out to a whole umber? No, you'll discover it's 27.5. Now divide 57 by 3. Is it a whole number? Yes, you will see that 57 divided by three is 19, which is indeed a whole number. Is 57 prime? No, 19 and 3 are its factors, which means the number is not a prime number, though its factor 19 is a prime number.

Divisibility and divisibility rules play a huge part in determining whether or not a number is prime. For instance, one divisibility rule states that if the number is even, it can be divided by two and is, therefore, not a prime number. Another helpful rule to remember is that if the added total of all the digits in a number is divisible by three, then the number itself is divisible by three and the number is not a prime number.

Similarly, if the last two digits of the number are divisible by 4, the entire number will be divisible by four and would therefore not be a prime number.

Other Methods and Helpful Hints for Determining Prime Numbers

Although it is not recommended to use until a student grasps the core concepts of prime numbers, the prime number calculator is a quick and easy method to determine if a number is prime or not, as are prime factorization trees, which is a method similar to factorization.

For factorization trees, one is usually expected to determine the common factors of multiple numbers. For instance, if one is factoring the number 30, he or she could begin with 10 x 3 or 15 x 2. In each case, the mathematician will continue to factor 10 (2 x 5) and 15 (3 x 5) and the end resulting prime factors will be the same: 2, 3 and 5 — after all, 5 x 3 x 2 = 30 as does 2 x 3 x 5.

Simple division with pencil and paper can also be a good method for teaching young learners how to determine prime numbers. First, take the number and try to divide it by two, then by three, four, and five if none of those division yield whole number results. Although this can be time-consuming and not particularly useful for large numbers, it's incredibly useful to help someone just starting out with the understanding of what makes a prime number prime. 

When working with prime numbers it's important that students know the difference between factors and multiples. These two terms are easily confused by learners, so it's important to emphasize that factors are numbers that can be divided evenly into the number being observed while multiples are the results of multiplying that number by another.