Find GCF and LCM | Varsity Tutors

SECTION 1

Where Did GCF and LCM Come From?

People have been breaking numbers into groups for thousands of years. Ancient builders needed to divide bricks into equal rows. Farmers needed to split crops into equal shares. These everyday problems led mathematicians to study factors (numbers that divide evenly into another number) and multiples (the results you get when you multiply a number by 1, 2, 3, and so on).

~300 BCE

Euclid's Algorithm

The Greek mathematician Euclid wrote a step-by-step method to find the greatest common factor of two numbers. His method still works today!

~500 CE

Indian Mathematicians

Aryabhata and other scholars in India used factors and multiples to solve problems with fractions and astronomy.

~800 CE

Al-Khwarizmi's Algebra

The Persian mathematician Al-Khwarizmi used common factors to simplify expressions, much like the distributive property you will learn here.

2010

Common Core Standards

The CCSS made GCF and LCM a key skill for 6th graders across the United States, connecting it to the distributive property.

So here is the big question this lesson answers: when you have two numbers, what is the largest number that divides into both? And what is the smallest number that both numbers divide into? These are the Greatest Common Factor (GCF) and the Least Common Multiple (LCM). Let's dig in!

SECTION 2

Core Principles & Definitions

Before we start finding GCF and LCM, you need to understand a few key ideas. These are the building blocks for everything else in this lesson.

Factor

A number that divides evenly into another number with no remainder. For example, 4 is a factor of 12 because 12 ÷ 4 = 3 with nothing left over.

Multiple

The result of multiplying a number by a whole number. The multiples of 5 are 5, 10, 15, 20, and so on.

Greatest Common Factor (GCF)

The largest factor that two (or more) numbers share. For 18 and 24, the GCF is 6 because 6 is the biggest number that goes into both.

Least Common Multiple (LCM)

The smallest multiple that two (or more) numbers share. For 4 and 6, the LCM is 12 because 12 is the first number that appears in both lists of multiples.

Distributive Property

A rule that says a × (b + c) = a × b + a × c. You can use the GCF to "factor out" a common number from a sum like 36 + 8 = 4 × (9 + 2).

✦ KEY TAKEAWAY

Think of GCF like splitting two different bags of candy into the biggest equal groups. If one bag has 18 pieces and the other has 24, the biggest group size that works for both bags is 6. LCM is like finding when two runners who start together at a track will both be at the starting line again — the first time their laps line up.

SECTION 3

Seeing GCF and LCM with Venn Diagrams

A great way to find the GCF and LCM is to use prime factorization (breaking a number down into prime numbers that multiply together) and then organize the prime factors in a Venn diagram. The diagram below shows how this works for the numbers 24 and 36.

The overlap (shared prime factors) gives the GCF of 12. Multiplying all the factors in the entire diagram gives the LCM of 72.

In the diagram, the left circle holds all the prime factors of 24 and the right circle holds all the prime factors of 36. The overlap in the middle shows the prime factors they share. Multiply the overlap to get the GCF. Multiply every factor in the whole diagram (without double-counting the overlap) to get the LCM.

SECTION 4

How to Find GCF and LCM Step by Step

Method 1: Listing Factors to Find GCF

To find the GCF of two numbers, list all the factors of each number. Then find the largest factor that appears in both lists. This works well for numbers up to 100.

GCF BY LISTING

Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 GCF(48, 60) = 12

The common factors are 1, 2, 3, 4, 6, and 12. The greatest of these is 12.

Method 2: Listing Multiples to Find LCM

To find the LCM of two numbers, list the multiples of each number until you find the first one they share. This is great for small numbers (up to 12).

LCM BY LISTING

Multiples of 4: 4, 8, 12, 16, 20, 24 … Multiples of 6: 6, 12, 18, 24, 30 … LCM(4, 6) = 12

Both lists share 12 and 24, but the least (smallest) common multiple is 12.

The Distributive Property Connection

Once you know the GCF, you can rewrite the sum of two numbers using the distributive property. Pull the GCF out front, and put the leftovers inside parentheses.

DISTRIBUTIVE PROPERTY WITH GCF

36 + 8 = 4 × (9 + 2)

The GCF of 36 and 8 is 4. Divide each number by 4: 36 ÷ 4 = 9, and 8 ÷ 4 = 2. So 36 + 8 = 4 × (9 + 2). Notice that 9 and 2 have no common factor other than 1.

SECTION 5

A Closer Look: Factor Trees and the Ladder Method

When numbers get bigger, listing every factor can be slow. Two faster tools are the factor tree and the ladder method (sometimes called the "birthday cake" method). The diagram below shows a factor tree for 60 and 84 side by side, then how to read the GCF.

Both factor trees end with green prime number circles. The primes shared by both trees are 2, 2, and 3. Multiply them to get GCF = 12.

In each factor tree, you start at the top with the original number. Then you split it into two factors. Keep splitting until every number at the bottom is a prime number (a number whose only factors are 1 and itself). Those green circles are the primes. Match up the primes that both trees share, and multiply them to get the GCF.

💡 Quick Tip: Ladder Method

Write both numbers side by side. Divide both by a common prime factor and write the results underneath, like floors of a building. Keep going until no prime divides both numbers. Multiply all the primes on the left side to get the GCF!

SECTION 6

Worked Example: GCF, LCM, and Distributive Property

Let's work through a full problem that combines all three skills: finding the GCF, finding the LCM, and using the distributive property.

Find the GCF of 54 and 72, then rewrite 54 + 72 using the distributive property. Also find the LCM of 6 and 8.

Step 1 — List the prime factorization of 54 and 72Use a factor tree or repeated division. 54 = 2 × 3 × 3 × 3. And 72 = 2 × 2 × 2 × 3 × 3.

54 = 2 × 3³ and 72 = 2³ × 3²

Step 2 — Find the GCF by identifying shared prime factorsBoth have at least one 2 and two 3s. So take the smaller power of each shared prime: one 2 and two 3s. Multiply: 2 × 3 × 3 = 18.

GCF(54, 72) = 18

Step 3 — Rewrite 54 + 72 using the distributive propertyDivide each number by the GCF. 54 ÷ 18 = 3. And 72 ÷ 18 = 4. So 54 + 72 = 18 × (3 + 4). Check: 3 and 4 share no common factor other than 1. ✓

54 + 72 = 18 × (3 + 4) = 18 × 7 = 126 ✓

Step 4 — Find the LCM of 6 and 8 by listing multiplesMultiples of 6: 6, 12, 18, 24, 30 … Multiples of 8: 8, 16, 24, 32 … The first number that shows up in both lists is 24.

LCM(6, 8) = 24

✅ Check Your Work

Always verify! For the distributive property, multiply back out: 18 × 3 = 54 and 18 × 4 = 72. The sum is 54 + 72 = 126, which matches 18 × 7 = 126. For LCM, make sure 24 ÷ 6 = 4 (whole number ✓) and 24 ÷ 8 = 3 (whole number ✓).

SECTION 7

Comparing Methods: Which Should You Use?

You now know several ways to find the GCF and LCM. Each method has strengths and weaknesses. The table below helps you pick the best tool for the job.

Comparison of GCF/LCM methods

Method	Best For	Watch Out
Listing Factors	GCF of small numbers (under 50). Easy to understand.	Gets slow for big numbers. Easy to miss a factor.
Listing Multiples	LCM of numbers up to 12. Quick and visual.	Lists get long for bigger numbers. Only practical for small numbers.
Prime Factorization	GCF or LCM of larger numbers (up to 100). Very reliable.	Takes more steps. You need to know your prime numbers.
Ladder (Cake) Method	GCF of two or even three numbers at once. Organized layout.	Can be confusing at first. Practice helps!

✦ KEY TAKEAWAY

Think of these methods like tools in a toolbox. A screwdriver is great for small screws, but you need a power drill for bigger jobs. Listing works for small numbers, and prime factorization is your power tool for bigger ones. The best math students know when to switch tools!

SECTION 8

Connection to Future Math

GCF and LCM are not just one-time skills. They show up again and again in math. Here is a preview of where you will use them next.

How GCF and LCM connect to future topics

What You Learned Now	Where It Goes Next
GCF of two numbers	Simplifying fractions — divide top and bottom by the GCF to get the simplest form.
LCM of two numbers	Adding and subtracting fractions with different denominators — the LCM becomes the common denominator.
Distributive property with GCF	Factoring expressions in algebra — pulling out common factors from terms like 6x + 12 = 6(x + 2).
Prime factorization	Number theory and cryptography — prime numbers are the basis of computer security!

Every time you simplify a fraction or find a common denominator, you are using the exact same skills you practiced in this lesson. Mastering GCF and LCM now makes all of those future topics much easier.

SECTION 9

Practice Problems

Try these five problems. They start simple and get harder. Write your answer first, then check it!

PROBLEM 1 — CONCEPTUAL

In your own words, explain the difference between a factor and a multiple. Give one example of each using the number 8.

PROBLEM 2 — BASIC CALCULATION

Find the GCF of 28 and 42.

PROBLEM 3 — INTERMEDIATE

Find the LCM of 9 and 12. Then find the GCF of 45 and 75 and use it to rewrite 45 + 75 with the distributive property.

PROBLEM 4 — APPLIED

A school is making gift bags. They have 48 pencils and 80 erasers. They want each bag to have the same number of pencils and the same number of erasers, with nothing left over. What is the greatest number of bags they can make? How many pencils and erasers go in each bag?

PROBLEM 5 — CRITICAL THINKING

True or false: For any two whole numbers, the product of the GCF and the LCM always equals the product of the two original numbers. Test this with the pair 8 and 12, and explain why it works.

SUMMARY

Lesson Summary

The Greatest Common Factor (GCF) is the largest number that divides evenly into two given numbers. You can find it by listing factors or by using prime factorization (breaking numbers into their prime building blocks and multiplying the shared ones). The Least Common Multiple (LCM) is the smallest number that both given numbers divide into evenly. For numbers up to 12, listing multiples is a quick way to find it.

Once you know the GCF of two numbers, you can use the distributive property to rewrite their sum. Pull the GCF out front and put the leftover numbers inside parentheses: for example, 36 + 8 = 4 × (9 + 2). This skill connects directly to simplifying fractions, finding common denominators, and factoring in algebra. Master these tools now, and they will help you for years to come!