Lesson 65: Prime Factorization

New Concept

When we write a composite number as a product of its prime factors, we have written the prime factorization of the number.

What’s next

Next, we will explore two powerful methods for finding prime factors—division by primes and factor trees—through worked examples and practice problems.

Property

When we write a composite number as a product of its prime factors, we have written the prime factorization of the number. For example, the prime factorization of 8 is $2 \cdot 2 \cdot 2$.

Examples

$12 = 2 \cdot 2 \cdot 3$
$20 = 2 \cdot 2 \cdot 5$
$45 = 3 \cdot 3 \cdot 5$

Explanation

Think of it as a secret recipe! Every whole number bigger than 1 is either prime or can be built by multiplying primes. Finding the prime factorization means discovering that unique set of prime number ingredients. You must break the number all the way down until only prime pieces are left, no composites allowed!

Property

To factor a number using division by primes, write the number in a division box and begin dividing by the smallest prime number that is a factor. Continue dividing the quotients by prime numbers until the final quotient is 1.

Examples

To factor 36: $36 \div 2 = 18$; $18 \div 2 = 9$; $9 \div 3 = 3$; $3 \div 3 = 1$. So, $36 = 2 \cdot 2 \cdot 3 \cdot 3$.
To factor 48: $48 \div 2 = 24$; $24 \div 2 = 12$; $12 \div 2 = 6$; $6 \div 2 = 3$; $3 \div 3 = 1$. So, $48 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3$.
To factor 50: $50 \div 2 = 25$; $25 \div 5 = 5$; $5 \div 5 = 1$. So, $50 = 2 \cdot 5 \cdot 5$.

Explanation

This handy method is like an upside-down division cake. You start with your number and repeatedly divide it by the smallest possible prime numbers. You stack the divisions until you reach a quotient of 1. All the prime divisors you used along the side are the secret ingredients that make up your number’s prime factorization.

Property

To make a factor tree, think of any two whole numbers whose product is the target number; these become the first branches. Continue factoring any composite branches until all the ends of the branches are prime numbers.

Examples

For 36, start with $4 \cdot 9$. Then $4 \to 2 \cdot 2$ and $9 \to 3 \cdot 3$. The primes are $2, 2, 3, 3$. So, $36 = 2 \cdot 2 \cdot 3 \cdot 3$.
For 40, start with $4 \cdot 10$. Then $4 \to 2 \cdot 2$ and $10 \to 2 \cdot 5$. The primes are $2, 2, 2, 5$. So, $40 = 2 \cdot 2 \cdot 2 \cdot 5$.
For 60, start with $6 \cdot 10$. Then $6 \to 2 \cdot 3$ and $10 \to 2 \cdot 5$. The primes are $2, 2, 3, 5$. So, $60 = 2 \cdot 2 \cdot 3 \cdot 5$.

Explanation

This is a visual way to hunt for primes! Start with your number as the trunk and split it into any two factors as branches. If a branch isn't a prime number, it's not a leaf yet—split it again! Keep branching out until every single branch tip is a prime number. Collect your prime 'leaves' for the answer.