Prime Factorization

SECTION 1

Where Did Prime Factorization Come From?

People have been fascinated by prime numbers for thousands of years. A prime number is a whole number greater than 1 that can only be divided evenly by 1 and itself. Ancient mathematicians realized that every number is built from these special primes, like how every LEGO creation is built from individual bricks.

The idea of breaking numbers into primes became one of the most powerful tools in all of mathematics. It helps us simplify fractions, find common denominators, and solve problems about sharing and grouping. Let's see how this idea developed over time.

~300 BC

Euclid Proves Infinite Primes

The Greek mathematician Euclid proved that there are infinitely many prime numbers. He also showed that every number can be written as a product of primes.

~200 BC

The Sieve of Eratosthenes

Eratosthenes invented a method to find all prime numbers up to any limit. You write out numbers and cross off multiples — the ones left standing are prime!

1801

Fundamental Theorem of Arithmetic

Carl Friedrich Gauss formally proved that every whole number greater than 1 has exactly one unique prime factorization. This is why we can rely on it every time.

Today

Primes Power the Internet

Modern encryption (the codes that keep your passwords safe online) depends on the fact that multiplying large primes is easy, but breaking a huge number back into primes is extremely hard for computers.

So here's the big question: how do we actually break a number into its prime pieces? And once we do, how does that help us find the Greatest Common Factor (GCF) and Least Common Multiple (LCM)? Let's find out.

SECTION 2

Core Principles & Definitions

Before we start factoring, let's make sure we understand the key vocabulary. These definitions are the foundation for everything else in this lesson.

Prime Number

A whole number greater than 1 whose only factors are 1 and itself. Examples: 2, 3, 5, 7, 11, 13. Notice that 2 is the only even prime.

Composite Number

A whole number greater than 1 that has more than two factors. For example, 12 is composite because 1, 2, 3, 4, 6, and 12 all divide into it evenly.

Writing a composite number as a product of only prime numbers. For example, 12 = 2 × 2 × 3, or 12 = 2² × 3.

Greatest Common Factor (GCF)

The largest number that divides evenly into two or more numbers. Also called the Greatest Common Divisor (GCD). The GCF of 12 and 18 is 6.

Least Common Multiple (LCM)

The smallest number that is a multiple of two or more numbers. The LCM of 4 and 6 is 12, because 12 is the first number that both 4 and 6 divide into evenly.

✦ KEY TAKEAWAY

Think of prime factorization like a recipe. Every composite number is a "dish" made from prime "ingredients." The number 60 is made from the primes 2, 2, 3, and 5 — just like a sandwich is made from bread, cheese, and lettuce. No matter how you take the sandwich apart, you always end up with the same ingredients. That's what makes prime factorization so reliable!

SECTION 3

The Factor Tree — A Visual Breakdown

A factor tree is a diagram that helps you break a number down step by step. You start with your number at the top. Then you split it into two factors. If a factor is composite, you split it again. You keep going until every branch ends in a prime number.

This factor tree shows how 60 splits into 6 × 10, then each of those splits further until we reach all primes: 2 × 2 × 3 × 5. The glowing circles at the bottom are the primes — your final answer.

Notice that no matter how you split 60 — you could start with 2 × 30, or 4 × 15 — you always end up with the same set of primes: 2, 2, 3, and 5. That's the beauty of the Fundamental Theorem of Arithmetic. Every number has one unique prime factorization.

💡 Tip: Where to Start?

When building a factor tree, start by checking the smallest primes first. Is the number even? Divide by 2. Does it end in 0 or 5? Divide by 5. Is the digit sum divisible by 3? Divide by 3. These quick checks make factoring faster!

SECTION 4

The Math Behind GCF and LCM

Once you have the prime factorization of two numbers, finding the GCF and LCM becomes a simple process. Here are the rules.

PRIME FACTORIZATION FORM

N = p₁ᵃ × p₂ᵇ × p₃ᶜ × …

Every composite number N can be written as a product of prime numbers (p₁, p₂, p₃, …) raised to powers (a, b, c, …). For example, 72 = 2³ × 3².

GREATEST COMMON FACTOR (GCF)

GCF = (shared primes) raised to the LOWEST power

Look at the prime factorizations of both numbers. Pick out only the primes they share. For each shared prime, use the smaller exponent. Multiply those together.

LEAST COMMON MULTIPLE (LCM)

LCM = (all primes) raised to the HIGHEST power

List every prime that appears in either number's factorization. For each prime, use the larger exponent. Multiply those together.

USEFUL RELATIONSHIP

GCF(a, b) × LCM(a, b) = a × b

This handy shortcut lets you check your work. If you multiply the GCF and LCM of two numbers, you should get the product of the two original numbers.

Here's the key memory trick: GCF = lowest powers, LCM = highest powers. The GCF is the biggest piece they already share. The LCM is the smallest number that fits both.

SECTION 5

GCF vs. LCM — A Side-by-Side Look

Students often mix up GCF and LCM. The diagram below shows both processes side by side using the numbers 36 and 48. Study how they compare.

The left box shows the GCF process (pick the lowest powers) and the right box shows the LCM process (pick the highest powers). The check at the bottom confirms our answers are correct.

Side-by-side steps for GCF vs. LCM

Step	GCF (Greatest Common Factor)	LCM (Least Common Multiple)
1. Find prime factorizations	Same for both — write each number as primes	Same for both — write each number as primes
2. Which primes?	Only shared primes	All primes from both numbers
3. Which exponent?	The smaller exponent	The larger exponent
4. Result	Multiply the chosen prime powers	Multiply the chosen prime powers

SECTION 6

Worked Example: Finding GCF and LCM of 54 and 90

Let's walk through a complete example. We'll find both the GCF and LCM of 54 and 90 using prime factorization.

Find the GCF and LCM of 54 and 90

Step 1 — Prime Factorize 5454 is even, so divide by 2: 54 ÷ 2 = 27. Now 27: is it divisible by 2? No. By 3? Yes! 27 ÷ 3 = 9. Then 9 ÷ 3 = 3. And 3 is already prime.

54 = 2 × 3 × 3 × 3 = 2¹ × 3³

Step 2 — Prime Factorize 9090 is even, so divide by 2: 90 ÷ 2 = 45. Then 45 ÷ 3 = 15. Then 15 ÷ 3 = 5. And 5 is prime.

90 = 2 × 3 × 3 × 5 = 2¹ × 3² × 5¹

Step 3 — Find the GCF (lowest powers of shared primes)Both numbers share the primes 2 and 3. For 2: both have 2¹, so we use 2¹. For 3: 54 has 3³ and 90 has 3², so we pick the lower one — 3². The prime 5 only appears in 90, so we skip it for GCF.

GCF = 2¹ × 3² = 2 × 9 = 18

Step 4 — Find the LCM (highest powers of all primes)List every prime from both: 2, 3, and 5. For 2: both have 2¹ → use 2¹. For 3: 54 has 3³ and 90 has 3², so we pick the higher one — 3³. For 5: only 90 has 5¹ → use 5¹.

LCM = 2¹ × 3³ × 5¹ = 2 × 27 × 5 = 270

Step 5 — Check your workUse the relationship GCF × LCM = a × b. That gives us 18 × 270 = 4,860. And 54 × 90 = 4,860. They match!

18 × 270 = 4,860 = 54 × 90 ✓ Confirmed!

SECTION 7

Comparing Methods for Finding GCF and LCM

Prime factorization isn't the only way to find GCF and LCM. You may have also learned listing factors or listing multiples. Let's compare all three methods so you can pick the best one for each situation.

Comparing three GCF/LCM methods

Method	Strengths	Limitations
Listing Factors/Multiples	Easy to understand. Great for small numbers like 12 and 18.	Slow for big numbers. Easy to miss a factor or get tired listing multiples.
Prime Factorization	Organized and reliable. Works well for large numbers. Gives both GCF and LCM from the same work.	Takes a few more steps for small, easy numbers. You need to know how to build factor trees.
Ladder (Cake) Method	Compact and visual. Finds GCF and LCM at the same time. Good for two or three numbers.	Less common in textbooks. Can be confusing if you haven't practiced it.

🔧 WHEN TO USE PRIME FACTORIZATION

Think of the listing method like searching every locker in a hallway to find your book — it works, but it's slow. Prime factorization is like having a master key that opens only the lockers you need. For bigger numbers (anything over about 30), prime factorization is almost always faster and more reliable.

SECTION 8

Where Does Prime Factorization Lead?

Prime factorization is a skill you'll use again and again in math. Let's see how what you've learned connects to topics you'll study later.

How today's skills connect to future math topics

What You Learned Now	Where It Goes Next
GCF of two numbers	Simplifying fractions (divide top and bottom by the GCF)
LCM of two numbers	Finding common denominators to add and subtract fractions
Writing numbers as prime products	Factoring algebraic expressions (like x² + 5x + 6) in Algebra 1
Understanding unique factorization	Cryptography and number theory in advanced math and computer science

In particular, every time you simplify a fraction like ⁴⁸⁄₇₂, you're really using the GCF. And every time you find a common denominator, you're using the LCM. Mastering prime factorization now gives you a superpower for all of those future problems.

🌟 Fun Fact

The largest known prime number (as of early 2025) has over 41 million digits! It was found by a worldwide volunteer computing project. People are still hunting for bigger ones.

SECTION 9

Practice Problems

Time to practice! Try each problem on your own before checking the answer. The problems get harder as you go.

PROBLEM 1 — CONCEPTUAL

Is the number 51 prime or composite? Explain how you know.

PROBLEM 2 — BASIC CALCULATION

Write the prime factorization of 120.

PROBLEM 3 — INTERMEDIATE

Find the GCF and LCM of 84 and 126 using prime factorization.

PROBLEM 4 — APPLIED

Maria has 48 red beads and 60 blue beads. She wants to make identical bracelets using all the beads, with each bracelet having the same number of red beads and the same number of blue beads. What is the greatest number of bracelets she can make?

PROBLEM 5 — CRITICAL THINKING

Two numbers have a GCF of 6 and an LCM of 180. One of the numbers is 36. What is the other number? (Hint: use the relationship GCF × LCM = a × b.)

SUMMARY

Lesson Summary

Every whole number greater than 1 is either prime (only divisible by 1 and itself) or composite (can be broken into smaller factors). Prime factorization means writing a composite number as a product of its prime building blocks. You can use a factor tree to find these primes step by step. Thanks to the Fundamental Theorem of Arithmetic, every number has exactly one unique prime factorization.

To find the Greatest Common Factor (GCF), compare the prime factorizations and multiply the shared primes using the lowest exponents. To find the Least Common Multiple (LCM), use all primes from both numbers with the highest exponents. You can always check your work with the relationship GCF × LCM = a × b. These skills are the foundation for simplifying fractions, finding common denominators, and solving real-world sharing and grouping problems.