English:Greatest Common Factors
Greatest Common Factors
Introduction
A greatest common factor, usually shortened to GCF, is the largest positive whole number that is a factor of two or more numbers. A factor divides a number exactly, with no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
You use GCFs when you want to split things into the largest possible equal groups, arrange objects in equal rows, tile rectangles with the largest possible square tiles, or simplify fractions. In some books, you may also see the names greatest common divisor or GCD. They mean the same idea.
By the end of this aiMOOC, you should be able to identify factors and common factors, find the GCF in several ways, explain why your answer is the greatest common factor, and use GCFs to solve real-life problems.
The chart above shows GCD values for small numbers. Look for patterns: when two numbers share many factors, their GCF can be larger; when they share only the factor 1, their GCF is 1.
Build the Idea Step by Step
What Is a Factor?
A factor of a number is a whole number that divides it exactly. One useful way to find factors is to make factor pairs.
For 18:
- 1 × 18 = 18
- 2 × 9 = 18
- 3 × 6 = 18
So the factors of 18 are 1, 2, 3, 6, 9, and 18.
Factors are different from multiples. Factors fit into a number. Multiples are made by multiplying a number by whole numbers. For example, 4 is a factor of 20, while 20 is a multiple of 4.
What Is a Common Factor?
A common factor is a factor shared by two or more numbers.
Compare 18 and 24:
Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Their common factors are 1, 2, 3, and 6. The greatest of these is 6, so GCF(18, 24) = 6.
A quick check is useful: 18 ÷ 6 = 3 and 24 ÷ 6 = 4. Because both answers are whole numbers, 6 is a common factor. Because no larger common factor exists, it is the GCF.
Method 1: List All Factors
Listing factors is a strong method when the numbers are not too large.
Try 20 and 30:
Factors of 20: 1, 2, 4, 5, 10, 20
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Common factors: 1, 2, 5, 10
Therefore, GCF(20, 30) = 10.
A careful strategy is to write factor pairs from the outside inward. Stop when the factors begin to repeat. This helps you avoid missing a factor.
Method 2: Use Prime Factorization
Every whole number greater than 1 can be written as a product of prime numbers. This is called prime factorization.
A factor tree is one visual way to break a composite number into prime factors.
For example:
36 = 2 × 2 × 3 × 3
48 = 2 × 2 × 2 × 2 × 3
The shared prime factors are 2, 2, and 3. Multiply them:
2 × 2 × 3 = 12
So GCF(36, 48) = 12.
Be careful not to use a prime factor more times than it appears in both numbers. The GCF uses only the prime factors the numbers share.
Method 3: Think with Equal Groups
GCF problems often hide inside sharing situations.
Suppose you have 18 red beads and 24 blue beads. You want to make the greatest possible number of identical bags without leftovers. Each bag must contain the same number of red beads and the same number of blue beads.
The number of bags must divide both 18 and 24. The greatest number that does this is 6. You can make 6 bags, with 3 red beads and 4 blue beads in each bag.
This is why the word common matters: the group count must work for every quantity.
Method 4: Think with Rectangles and Square Tiles
GCFs also appear in geometry. Imagine a 24 by 60 rectangle. You want to cover it with identical square tiles, with no gaps and no cutting. The side length of each square must divide both 24 and 60.
The largest possible square has side length 12 because 12 is the GCF of 24 and 60.
This picture gives a visual meaning to the GCF: it is the side length of the largest square tile that fits exactly along both dimensions.
Useful Connections
GCF and Fractions
The GCF helps you simplify fractions efficiently. Consider 18/24.
The GCF of 18 and 24 is 6. Divide the numerator and denominator by 6:
18 ÷ 6 = 3
24 ÷ 6 = 4
So 18/24 simplifies to 3/4.
When you divide both parts of a fraction by their GCF, the resulting fraction is in simplest form.
GCF and Coprime Numbers
Two numbers are coprime when their GCF is 1. This does not mean both numbers must be prime.
For example, 8 and 15 are not both prime, but their only common factor is 1. Therefore, GCF(8, 15) = 1.
Recognizing coprime numbers can help you see quickly when a fraction is already in simplest form.
GCF and LCM Are Different
The GCF asks for the greatest factor shared by numbers. The least common multiple, or LCM, asks for the smallest positive multiple shared by numbers.
For 6 and 8:
GCF(6, 8) = 2
LCM(6, 8) = 24
A useful question is: Am I dividing into equal groups, or am I looking for when repeating patterns meet? Equal grouping often suggests GCF. Repeating cycles often suggest LCM.
Problem-Solving Strategies
How to Recognize a GCF Problem
A problem may involve a GCF when you see ideas such as:
- making the greatest possible number of equal groups;
- dividing several quantities with no leftovers;
- cutting equal pieces as large as possible;
- arranging items into identical sets;
- finding the largest square tile that fits two dimensions exactly.
The word greatest does not always appear in the problem. You need to think about what is being maximized.
A Four-Step Plan
- Identify the numbers that must be divided evenly.
- Find common factors by listing factors, using prime factorization, or reasoning from known facts.
- Choose the greatest common factor.
- Check that the GCF divides every original number exactly and that it fits the situation.
Example: A teacher has 28 pencils and 42 erasers. She wants to make the greatest possible number of identical prize packs with no leftovers.
The common factors of 28 and 42 are 1, 2, 7, and 14. The GCF is 14. Therefore, she can make 14 packs. Each pack has 2 pencils and 3 erasers.
Common Mistakes to Avoid
Mistake 1: Choosing any common factor. If 2 divides both numbers but 6 also divides both, then 2 is not the greatest common factor.
Mistake 2: Mixing up factors and multiples. Factors divide a number; multiples are results of multiplying a number.
Mistake 3: Stopping factor lists too early. Check every factor pair so that you do not miss a larger common factor.
Mistake 4: Multiplying every prime factor you see. In prime factorization, use only factors shared by all the numbers.
Mistake 5: Ignoring the context. If a word problem asks for the greatest number of equal groups, your final answer describes a number of groups, not necessarily the number of objects in each group.
Challenge Corner: Euclid's Algorithm
For larger numbers, mathematicians often use the Euclidean algorithm. This method repeatedly replaces a pair of numbers with a smaller pair while keeping the same GCF.
For example, to find GCF(48, 18):
48 = 2 × 18 + 12
18 = 1 × 12 + 6
12 = 2 × 6 + 0
The last nonzero remainder is 6, so GCF(48, 18) = 6.
You do not need this method for every Grade 5–6 problem, but it shows that there are efficient ways to find GCFs without listing every factor.
Interactive Tasks
Quiz: Test Your Knowledge
What is the greatest common factor of 12 and 18? (6) (!2) (!3) (!36)
Which number is a factor of 24? (6) (!5) (!7) (!10)
What are all the common factors of 8 and 12? (1, 2, and 4) (!1 and 2) (!2, 4, and 8) (!4, 8, and 12)
Which method is especially useful for finding a GCF of small numbers? (List all factors) (!List all multiples) (!Add the numbers) (!Subtract one number once)
Which is the prime factorization of 18? (2 × 3 × 3) (!2 × 9) (!3 × 6) (!1 × 18)
What is the GCF of two coprime numbers? (1) (!0) (!2) (!The larger number)
You have 18 red beads and 24 blue beads and want the greatest number of identical bags with no leftovers. How many bags can you make? (6) (!3) (!4) (!42)
What is the side length of the largest square tile that exactly covers a 24 by 60 rectangle? (12) (!6) (!24) (!60)
What is the GCF of 15 and 25? (5) (!3) (!10) (!75)
Which statement correctly compares factors and multiples? (Factors divide a number exactly) (!Factors are always larger than a number) (!Multiples divide a number exactly) (!Multiples are always prime)
Memory Game
| Factor | A whole number that divides another number exactly |
| Common factor | A factor shared by two or more numbers |
| Greatest common factor | The largest positive factor shared by all given numbers |
| Prime number | A whole number greater than one with exactly two positive factors |
| Prime factorization | Writing a number as a product of prime numbers |
| Coprime numbers | Two numbers whose greatest common factor is one |
| Factor pair | Two whole numbers multiplied to make a given product |
Drag and Drop
| Match the correct terms. | Topic |
|---|---|
| Greatest common factor | Largest factor shared by all given numbers |
| Factor | Whole number that divides another number exactly |
| Common factor | Factor that belongs to more than one number |
| Prime factorization | Product of prime numbers equal to the original number |
| Coprime numbers | Pair of numbers with greatest common factor one |
Match each mathematical term with the explanation that belongs to it. Then create one example for each match in your notebook.
Crossword Puzzle
| Factor | What whole-number term divides another number exactly? |
| Common | What word describes a factor shared by more than one number? |
| Greatest | What word in GCF tells you to choose the largest shared factor? |
| Prime | What type of number has exactly two positive factors? |
| Coprime | What describes two numbers whose GCF is one? |
| Factorization | What process writes a number as a product of factors? |
LearningApps
Cloze Text
Open-Ended Tasks
Easy
- Factor Hunt: Choose five numbers from 10 to 50, list every factor of each number, and circle the numbers that have more than six factors.
- GCF Card Sort: Make ten number cards, pair the cards in five different ways, and find the GCF of each pair using factor lists.
- Equal Group Drawing: Draw 18 circles and 24 squares, arrange them into the greatest possible number of identical groups, and explain how your picture shows the GCF.
- Math Vocabulary Poster: Create a one-page poster that explains factor, common factor, greatest common factor, prime number, and coprime using your own examples.
Standard
- Factor Tree Gallery: Build factor trees for 24, 36, 42, and 60, write each prime factorization, and use two trees to demonstrate how shared prime factors produce a GCF.
- Classroom Pack Problem: Invent a story about two supplies that must be divided into the greatest possible number of equal packs, solve it, and exchange it with a classmate for checking.
- Square Tile Investigation: Draw rectangles with dimensions 12 by 18, 16 by 24, and 20 by 30, determine the largest square tile that fits each rectangle exactly, and connect each tile size to a GCF.
- GCF Interview: Interview a teacher, family member, craft worker, cook, or organizer about a situation involving equal groups or equal-size pieces, then explain whether a GCF could help.
Advanced
- Method Comparison: Find the GCF of five number pairs using both factor lists and prime factorization, record the steps, and write which method you prefer for different sizes of numbers.
- Design a GCF Game: Create a board, card, or digital game in which players earn points by identifying common factors and GCFs, then test the rules with at least two players and revise them.
- GCF Video Lesson: Produce a two- to four-minute teaching video that explains one GCF method, includes a worked example and a word problem, and ends with a challenge for viewers.
- Euclidean Algorithm Investigation: Use repeated division to find the GCF of at least four pairs of larger numbers, compare the results with another method, and describe what stays the same during the process.
Learning Assessment
- Explain and Justify: Find GCF(36, 54) and write a short explanation showing why your answer is common to both numbers and why no greater common factor is possible.
- Choose a Strategy: Find GCF(84, 126) using a method you consider efficient, then explain why you chose that method instead of another one.
- Apply to Sharing: A club has 48 juice boxes and 60 snack bars and wants the greatest possible number of identical sets with no leftovers; determine the number of sets and the contents of each set.
- Apply to Geometry: A rectangular floor is 30 units by 42 units and will be covered by the largest possible identical square tiles; determine the tile side length and explain the connection to GCF.
- Analyze an Error: A student says the GCF of 18 and 30 is 3 because 3 divides both numbers; identify the mistake, correct the answer, and show evidence.
- Transfer to Fractions: Simplify 42/56 by using a GCF, then explain why dividing by the GCF guarantees that the final fraction is in simplest form.
Evidence of Learning
Strong evidence of learning includes both correct answers and clear reasoning. You should be able to show the following:
Knowledge: You can define factor, common factor, greatest common factor, prime factorization, and coprime numbers, and you can explain the difference between factors and multiples.
Skills: You can find all factors of a number, identify shared factors, calculate a GCF by listing factors or using prime factorization, and check your result by division.
Problem solving: You can decide when a real-life situation requires a GCF and explain what the GCF means in that context.
Representations: You can use factor lists, factor trees, equal groups, arrays, rectangles, or square tiling to make GCF reasoning visible.
Products: Your posters, diagrams, games, written solutions, investigations, or videos accurately communicate GCF ideas to another learner.
Transfer: You can use GCF reasoning to simplify fractions, organize equal groups, and solve unfamiliar sharing or measurement problems.
OERs on the Topic
The English Wikipedia article uses the name greatest common divisor, which is another name for greatest common factor.
Linked Learning Areas
Greatest common factors connect number sense, multiplication, division, prime numbers, fractions, geometry, and problem solving. Understanding these links helps you choose a method instead of memorizing one procedure.