Prime and Composite Numbers

To understand prime and composite numbers, we first need to know about factors. Factors are the numbers you multiply together to get another number. For example, $2$ and $3$ are factors of $6$ because $2 \times 3 = 6$. The numbers $1$ and $6$ are also factors of $6$ because $1 \times 6 = 6$.

Based on how many factors a number has, we can sort whole numbers into two main categories: prime and composite.

Prime Numbers

A prime number is a whole number greater than $1$ that has exactly two factors: $1$ and the number itself.

If a number is prime, the only way to make it by multiplying two whole numbers is to multiply $1$ by the number itself.

• Example: $7$ is a prime number. The only factors of $7$ are $1$ and $7$ ($1 \times 7 = 7$).
• Example: $13$ is a prime number. Its only factors are $1$ and $13$.

Fun Fact: $2$ is the smallest prime number, and it is the only even prime number! Every other even number can be divided by $2$, which gives it more than two factors.

Composite Numbers

A composite number is a whole number that has more than two factors.

If a number is composite, you can arrange it into different rectangular arrays. It can be divided evenly by numbers other than $1$ and itself.

• Example: $8$ is a composite number. Its factors are $1, 2, 4,$ and $8$ (because $1 \times 8 = 8$ and $2 \times 4 = 8$).
• Example: $15$ is a composite number. Its factors are $1, 3, 5,$ and $15$.

The Special Number: 1

What about the number $1$? The number $1$ only has one factor: itself. Because it does not have exactly two factors, and it does not have more than two factors, the number $1$ is neither prime nor composite.

Example Problems

Let's apply what we've learned to some examples.

Is 29 prime or composite?

Let's look for the factors of $29$. Are there any numbers besides $1$ and $29$ that multiply together to make $29$? It cannot be divided evenly by $2, 3, 4, 5,$ or any other number. The only factors are $1$ and $29$. Answer: $29$ is prime.

Is 51 prime or composite?

Let's find the factors of $51$. We know $1 \times 51 = 51$. Are there any others? If we use the divisibility rule for $3$ (add the digits: $5 + 1 = 6$, which is divisible by $3$), we know $3$ goes into $51$. $3 \times 17 = 51$. The factors of $51$ are $1, 3, 17,$ and $51$. Since it has more than two factors, it is composite. Answer: $51$ is composite.

List all prime numbers less than 20

Let's count up from $2$ and check each number:

• $2$ (Prime: $1, 2$)
• $3$ (Prime: $1, 3$)
• $4$ (Composite: $1, 2, 4$)
• $5$ (Prime: $1, 5$)
• $6$ (Composite: $1, 2, 3, 6$)
• $7$ (Prime: $1, 7$)
• $8, 9, 10$ (Composite)
• $11$ (Prime: $1, 11$)
• $12$ (Composite)
• $13$ (Prime: $1, 13$)
• $14, 15, 16$ (Composite)
• $17$ (Prime: $1, 17$)
• $18$ (Composite)
• $19$ (Prime: $1, 19$)

Answer: The prime numbers less than $20$ are: $2, 3, 5, 7, 11, 13, 17,$ and $19$.