Prime And Composite Numbers – Lesson

Earlier on, it was mentioned that factors are very important. But why?

One of the reasons was because they divide the natural numbers into 2 categories based on the number of factors that a number has. More specifically, the number of positive factors that a number has!

These 2 categories are called Prime and Composite numbers. Let’s start with prime numbers. Prime numbers are numbers that have exactly 2 positive factors!

Now, we know that 1 is a factor of all numbers. We also know that a number is a factor of itself(for example, 6 is a factor of 6). But, that is already 2 factors! That means a prime number has no factors other than 1 or itself. This means that the only product pair, for a prime “k”, is $$1\cdot{k}=k$$ We will see a bit later in this lesson why this is so important.

A lot of the numbers that we deal with tend to have more than 2 factors. As you have guessed, these numbers are called composite numbers! Composite numbers are natural numbers with more than 2 positive factors. Thus, unlike prime numbers, they can be written as the product of 2 numbers other than itself and 1. For example, since we can write $$2\cdot3=6,$$ We know that 6 is not a prime number but rather a composite number. In essence, composite numbers must be divisible by a number other than 1 and itself. This also means that a composite number is divisible by another/other prime(s) number(s)(think about why this is true).

For example, the number 24 = 6 × 4. Both 6 and 4 are not prime… but rewriting them in a similar way, we get 6 = 2 × 3 and 4 = 2 × 2. Both 2 and 3 ARE prime numbers! Doing this for any composite numbers, we will eventually get to primes. We cannot follow a similar process for prime numbers. This is why, prime numbers are considered to be the building blocks of Number Theory. All natural numbers can be built as a product of them. This is referred to Prime Factorization, our next topic!

Example: Is 8 composite or prime?

Answer: Since 8 is divisible by 2, it has a factor of 2, which means it has at least 3 factors(try to calculate how many factors it does have!)

Example: Is 1 composite or prime?

Answer: The number 1 only has 1 positive factor: 1. Thus, it does not meet the requirements to be prime(exactly 2 factors) or composite(more than 2 factors). This means that 1 is neither prime nor composite.

Example: List all the prime numbers from 1-25.

Answer: We just individually test the numbers. Obviously, we can use some shortcuts to help us out! All even numbers other than 2 are automatically composite since they are divisible by 2. Similarly, we can do this for 3, 5, 7, etc. Thus, the prime numbers from 1-25 are: $$2, 3, 5, 7, 11, 13, 17, 19, 23$$

Let’s now see the video for this lesson!