Lesson 21: Prime and Composite Numbers, Prime Factorization

New Concept

Counting numbers greater than $1$ are either prime or composite. A prime number has exactly two factors, while a composite number has more than two.

A prime number is a counting number greater than $1$ whose only factors are $1$ and the number itself.

What’s next

This is a foundational concept. Next, you'll master two methods for finding prime factorizations and apply this skill to solve problems involving the GCF.

Property

A counting number greater than $1$ is either prime or composite. A prime number has exactly two different factors: $1$ and itself. A composite number has three or more factors. The number $1$ is neither prime nor composite.

Examples

The number $13$ is prime because its only factors are $1$ and $13$.
The number $18$ is composite because its factors are $1, 2, 3, 6, 9,$ and $18$.
The number $1$ is neither prime nor composite, as it has only one factor.

Explanation

Think of prime numbers as exclusive clubs with only two members allowed: the number itself and $1$. Composite numbers are like giant parties where anyone who is a factor can join in the fun. And poor number $1$? It’s a club of one, so it doesn't fit in either group, making it truly unique in the number world!

Property

Every composite number can be composed by multiplying two or more prime numbers. When we write a composite number as a product of prime numbers, we are writing the prime factorization of the number.

Examples

The prime factorization of $12$ is $2 \cdot 2 \cdot 3$, not $2 \cdot 6$ because $6$ is not a prime number.
The prime factorization of $49$ is $7 \cdot 7$.
The prime factorization of $100$ is $2 \cdot 2 \cdot 5 \cdot 5$.

Explanation

Prime factorization is like finding the secret recipe for a number. You break it down into its most basic, prime ingredients. No matter how you chop it up, you'll always end up with the same set of prime numbers. For example, the number $12$ will always be made of two $2$s and one $3$ in its prime recipe.

Property

A factor tree is a method to find the prime factorization of a number. Start by writing any two factors below the number. If a factor is not prime, continue breaking it down into two new factors until all the 'branches' of the tree end in prime numbers.

Examples

To factor $54$, branch into $6$ and $9$. Then $6$ branches into $2$ and $3$, while $9$ branches into $3$ and $3$. The prime factors are $2, 3, 3, 3$.
To factor $28$, branch into $4$ and $7$. The $7$ is prime, but the $4$ branches into $2$ and $2$. The prime factors are $2, 2, 7$.

Explanation

Imagine a number is the trunk of a tree. It splits into two branches (factors). If a branch is composite, it splits again! You keep splitting the branches until they can’t split anymore, leaving you with the prime 'leaves' of the tree. These leaves, when multiplied together, give you the original trunk number. It's a visual way to decompose numbers.