A prime number is a numeral that is greater than 1 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.

### Factors vs. Multiples

When working with prime numbers, students should know the difference between factors and multiples. These two terms are easily confused, but **factors** are numbers that can be divided evenly into the given number, while **multiples** are the results of multiplying that number by another.

Additionally, prime numbers are whole numbers that must be greater than one, and as a result, zero and one are not considered prime numbers, nor is any number less than zero; the number two is the first prime number, as it can only be divided by itself and the number 1.

### Using Factorization

Using a process called factorization, mathematicians can quickly determine whether a number is prime. To use factorization, you need to know that 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. This is expressed in the prime factors of 10 as 5 and 2 since both 1 and 10 are not prime numbers.

An easy way for students to use factorization to determine if a number is prime is by giving them concrete counting items like beans, buttons, or coins. They can use these to divide objects into ever-smaller groups. For example, they could divide 10 marbles into two groups of five or five groups of two.

### Using a Calculator

After using the concrete method as described in the previous section, students can use calculators and the concept of divisibility to determine whether a number is prime.

Have students take a calculator and key in the number to determine whether it is prime. The number should divide into a whole number. For example, take the number 57. Have students divide the number by 2. They will see that the quotient is 27.5, which is not an even number. Now have them divide 57 by 3. They will see that this quotient is a whole number: 19. So, 19 and 3 are factors of 57, which is, then, not a prime number.

### Other Methods

Another way to find if a number is prime is by using a factorization tree, where students determine the common factors of multiple numbers. For instance, if a student is factoring the number 30, she could begin with 10 x 3 or 15 x 2. In each case, she continues to factor—10 (2 x 5) and 15 (3 x 5). The end result will yield the same prime factors: 2, 3 and 5 because 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, divide the number by two, then by three, four, and five if none of those factors yields a whole number. This method is useful to help someone just starting out to understand what makes a number prime.