Prime Factorization

🔢 Prime Factorization

Part 1 of 5 — Concept Introduction

Every whole number bigger than 1 is built out of special "building-block" numbers called primes. In this lesson you'll learn how to take any number apart into its prime building blocks — a skill called prime factorization.

But first, we need to know the difference between prime and composite numbers.

Prime Numbers

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

That means you cannot split a prime evenly into smaller equal groups (other than 1 group of itself).

The first prime numbers are:

$2,\ 3,\ 5,\ 7,\ 11,\ 13,\ 17,\ 19,\ 23,\ 29 \dots$

Fun fact: $2$ is the only even prime number. Every other even number can be divided by 2, so it has more than two factors! 🤯

Composite Numbers

A composite number is a whole number greater than 1 that has more than two factors. It can be built by multiplying smaller numbers together.

Examples: $4,\ 6,\ 8,\ 9,\ 10,\ 12,\ 14,\ 15,\ 16 \dots$

For example, $12$ has the factors $1, 2, 3, 4, 6, 12$ — that's six factors, way more than two, so $12$ is composite.

Important: The number $1$ is neither prime nor composite — it has only one factor (itself).

What Is Prime Factorization?

Prime factorization means writing a number as a product of prime numbers only.

It's like finding the exact recipe of prime building blocks that multiply together to make your number.

Example: Let's break apart $24$.

$24 = 2 \times 2 \times 2 \times 3$

Every factor on the right ($2, 2, 2, 3$) is prime — perfect! When a prime repeats, we can use an exponent to write it more neatly:

$24 = 2^3 \times 3$

The little raised $3$ means "$2$ is used three times." Both forms are correct and mean exactly the same thing. ✅

Concept Check 🎯

Make sure you can tell primes and composites apart.

📝 Worked Examples

Part 2 of 5 — Worked Examples

There are two friendly ways to find a prime factorization. Let's walk through both.

Method 1: The Factor Tree 🌳

Keep splitting a number into two factors until every branch ends in a prime.

Example: Factor $36$.

• $36 = 4 \times 9$
• Split $4 = 2 \times 2$ (both prime, stop ✅)
• Split $9 = 3 \times 3$ (both prime, stop ✅)

Collect every prime at the end of a branch:

$36 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2$

Tip: It doesn't matter how you split at the start. $36 = 6 \times 6$ gives the same prime factorization. Cool, right? 🌟

Method 2: The Division Ladder 🪜

Divide by the smallest prime that fits, again and again, until you reach $1$.

Example: Factor $60$.

The prime numbers you divided by are $2, 2, 3, 5$. So:

$60 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3 \times 5$

Check it: $2 \times 2 \times 3 \times 5 = 4 \times 15 = 60$ ✅

Your Turn 🧮

Use the division ladder to factor $\mathbf{18}$, step by step.

1. Divide $18$ by the smallest prime, $2$. What do you get?

2. Now divide that result by the prime $3$. What do you get?

3. Divide that result by the prime $3$ again. What do you get? (You should reach the number that ends the ladder.)

🧭 Guided Practice

Part 3 of 5 — Guided Practice

Practice finding full prime factorizations. Remember: the answer should contain only prime numbers.

Fill In the Factorization 🔍

Complete each prime factorization by choosing the correct value.

🌍 Application & Word Problems

Part 4 of 5 — Real-World Uses

Why bother breaking numbers into primes? Because prime factorization is a secret tool that helps you:

• Find the Greatest Common Factor (GCF) of two numbers
• Find the Least Common Multiple (LCM) of two numbers
• Simplify fractions quickly

Finding the GCF With Primes 🎒

Suppose you have $12$ pencils and $18$ erasers, and you want to make identical gift bags with no leftovers. The biggest number of equal bags is the GCF of $12$ and $18$.

Step 1 — Factor each number:

$12 = 2 \times 2 \times 3 \qquad 18 = 2 \times 3 \times 3$

Step 2 — Circle the primes they share: both have one $2$ and one $3$.

Step 3 — Multiply the shared primes:

$\text{GCF} = 2 \times 3 = 6$

So you can make 6 identical bags (each with 2 pencils and 3 erasers). 🎉

Word Problem Practice 🧮

A baker has $24$ muffins and $30$ cookies. She wants to fill identical boxes using all the treats with none left over. To find the most boxes, she finds the GCF of $24$ and $30$.

1. Find the prime factorization of $24$. Type the number of times the prime $2$ appears. (For $24 = 2 \times 2 \times 2 \times 3$, type how many $2$'s.)

2. The primes that $24$ and $30$ share are one $2$ and one $3$. Multiply them: $2 \times 3 = ?$

3. So what is the greatest number of identical boxes she can make? (It equals the GCF.)

Word Problem Check 📋

🏆 Review & Challenge

Part 5 of 5 — Review & Challenge

Awesome work! Here's everything you learned, all in one place.

Quick Summary Table

Remember:

• $2$ is the only even prime.
• The number $1$ is neither prime nor composite.
• Use exponents to write repeated primes neatly, like $2 \times 2 \times 2 = 2^3$.
• Both methods give the same prime factorization — every number has exactly one!

Now finish strong with these mixed challenge questions! 💪

Challenge Round 🎯

These mix all the ideas together. Take your time and check each answer!