You have reached the final lesson of **Foundations of Factors and Multiples ** — and it is the one that gives the course its name. Over the first three lessons, you mastered factors, factor pairs, complete factor lists, and multiples. Each of those skills was a stepping stone leading here.

Now we put them to work. In this lesson, we will sort every whole number greater than $1$ into one of two categories — prime or composite — based on how many factors it has. We will also settle the question of where $1$ belongs and discover why prime numbers earn the title of "building blocks" for all other numbers.

Recall from Lesson 2 that every whole number has a complete factor list. Some lists are short and some are long. Let's line up a few familiar examples:

A prime number is a whole number greater than $1$ that has exactly two factors: $1$ and the number itself. No other whole number divides it evenly.

Consider $13$. If we test every whole number from $2$ up to $12$, none of them divides $13$ without a remainder. The only factors are and , giving us exactly two, so is prime. The first several prime numbers are:

A composite number is a whole number greater than $1$ that has more than two factors. That means at least one whole number besides $1$ and the number itself divides it evenly.

Take $18$. Its complete factor list is $1, 2, 3, 6, 9, 18$ — six factors in total. Since that is more than two, is composite. Here are a few more examples:

The number $1$ is a common source of confusion. It might feel like a prime because it cannot be broken into smaller whole-number factors. However, recall the definition: a prime must have exactly two factors. The only factor of $1$ is $1$ itself, so its factor count is just one.

Because $1$ has fewer than two factors, it does not qualify as prime. And because it does not have more than two factors, it is not composite either. Mathematicians therefore classify $1$ as neither prime nor composite — a special case we will revisit at the end of the lesson when we see why this choice matters.

For small numbers like $2$ through $10$, you can probably classify them from memory. But what about a number like $31$ or $39$? The technique is the same systematic approach from Lesson 2: test potential factors starting from $2$.

Let's check $39$. We try : since is odd, does not divide it. We try : with no remainder. We found a factor other than and , so is — no need to keep testing.

Here is the most important idea in this lesson. Every composite number can be written as a product of prime numbers. Let's see this in action with $60$:

The factor $30$ is still composite, so we split it further:

In this lesson, you classified every whole number greater than $1$ as prime (exactly two factors) or composite (more than two factors) and established that $1$ is neither. You also discovered the deeper reason primes matter: they are the irreducible pieces from which every composite number is built, a principle captured by the Fundamental Theorem of Arithmetic.

Now it is time to put your understanding to the test. In the practice exercises ahead, you will classify numbers on sight, sort them into prime and composite groups, hunt for primes in unfamiliar ranges, and break a composite number all the way down to its prime building blocks. Jump in and see how solid your number sense has become!