How to determine whether a number is a prime or composite number

What is the difference between a prime and composite number?

You’ve already learned several ways to classify a whole number: as positive, negative, or $0$ ; as the opposite of a number, in terms of the whole numbers by which it’s divisible, and in terms of its multiples.

Another way to classify a whole number greater than $1$ is as either a “prime” number or a “composite” number. A whole number will either be a prime number or a composite number, but it can’t be both.

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A prime number is a whole number greater than $1$ that’s divisible by only $1$ and itself. We already know that every whole number is divisible by $1$ , because if you divide any whole number by $1$ , you’ll get the original number as the result. Here are some examples:

$10\div1=10$

$7\div1=7$

$316\div1=316$

And every whole number other than $0$ is divisible by itself, because if you divide any whole number other than $0$ by itself, you’ll get $1$ (a whole number) as the result. Here are some examples:

$10\div10=1$

$7\div7=1$

$316\div316=1$

Prime numbers are special because they are divisible by only $1$ and themselves. As an example, $7$ is a prime number because it’s divisible by $1$ ( $7\div1=7$ ) and by itself ( $7\div7=1$ ), but not by anything else. It’s not divisible by $2$ , $3$ , $4$ , $5$ , or $6$ , because none of these numbers divides evenly into $7$ .

The number $11$ is also a prime number, because the only numbers that divide evenly into $11$ are $1$ and $11$ .

Because $7$ and $11$ are divisible only by $1$ and themselves, we call them “prime” numbers.

Contrast this with composite numbers, which by definition are numbers greater than $1$ that are divisible by something other than just $1$ and themselves. For example, $6$ is a composite number; it’s divisible by $1$ ( $6\div1=6$ ) and by itself ( $6\div6=1$ ), but it’s also divisible by $2$ ( $6\div2=3$ ) and by $3$ ( $6\div3=2$ ). Because $6$ is divisible by more than just $1$ and itself, we call it a “composite” number.

How to determine whether a number is a prime number or a composite number

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Say whether the number is a prime number or composite number

Example

Say whether $21$ is prime or composite.

The number $21$ is divisible by $1$ ( $21\div1=21$ ) and by itself ( $21\div21=1$ ), but it’s also divisible by $3$ ( $21\div3=7$ ) and by $7$ ( $21\div7=3$ ). Because $21$ is divisible by more than just $1$ and itself, it’s a composite number, not a prime number.