Factors and Multiples: A Quick Review

The greatest common factor (GCF) and least common multiple (LCM) are based on two ideas we have already studied: factors and multiples.

A factor divides a number exactly.

For example, the factors of 12 are:

1, 2, 3, 4, 6, 12

A multiple is produced by multiplying a number by whole numbers.

Multiples of 12 include:

12, 24, 36, 48, 60, ...

GCF looks for factors that numbers share.

LCM looks for multiples that numbers share.

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What Is a Common Factor?

A common factor is a factor shared by two or more numbers.

Consider:

12 and 18

Factors of 12:

1, 2, 3, 4, 6, 12

Factors of 18:

1, 2, 3, 6, 9, 18

Common factors:

1, 2, 3, 6

The greatest of these common factors is:

6

Therefore:

GCF(12, 18) = 6

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What Is the Greatest Common Factor?

The greatest common factor (GCF) is the largest positive factor shared by two or more numbers.

It may also be called:

• greatest common divisor (GCD)
• highest common factor (HCF)

These terms describe the same basic idea.

For example:

GCF(20, 30) = 10

because 10 is the largest number that divides both 20 and 30 exactly.

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Finding the GCF by Listing Factors

One method is to list all the factors.

Find:

GCF(24, 36)

Factors of 24:

1, 2, 3, 4, 6, 8, 12, 24

Factors of 36:

1, 2, 3, 4, 6, 9, 12, 18, 36

Common factors:

1, 2, 3, 4, 6, 12

The greatest common factor is:

12

Therefore:

GCF(24, 36) = 12

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Checking a GCF

If 12 is the GCF of 24 and 36, it must divide both numbers exactly.

Check:

24 ÷ 12 = 2

36 ÷ 12 = 3

Both quotients are whole numbers.

Therefore, 12 is a common factor.

Since there is no larger common factor:

GCF = 12

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What Is a Common Multiple?

A common multiple is a multiple shared by two or more numbers.

Consider:

4 and 6

Multiples of 4:

4, 8, 12, 16, 20, 24, 28, 32, 36, ...

Multiples of 6:

6, 12, 18, 24, 30, 36, ...

Common multiples include:

12, 24, 36, ...

The smallest positive common multiple is:

12

Therefore:

LCM(4, 6) = 12

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What Is the Least Common Multiple?

The least common multiple (LCM) is the smallest positive multiple shared by two or more numbers.

For example:

Multiples of 5:

5, 10, 15, 20, 25, 30, ...

Multiples of 6:

6, 12, 18, 24, 30, ...

The first positive number appearing in both lists is:

30

Therefore:

LCM(5, 6) = 30

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Finding the LCM Using a Number Line

Multiples can also be represented as equal jumps on a number line.

For example, compare multiples of 4 and 6:

The first positive landing shared by both sequences gives the LCM.

This helps show that the LCM is not just a calculation—it is the first point at which two repeating patterns meet.

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GCF and LCM Are Different

GCF asks:

What is the largest factor the numbers share?

LCM asks:

What is the smallest positive multiple the numbers share?

For:

12 and 18

GCF:

6

LCM:

36

Do not confuse the two ideas.

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A Useful Way to Remember

Think:

GCF → factors → divide

LCM → multiples → multiply/repeat

GCF problems often involve:

• dividing objects into equal groups
• creating identical packages
• cutting materials into equal largest pieces

LCM problems often involve:

• repeating events
• schedules
• cycles
• events occurring together again

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Finding the GCF of Three Numbers

Find:

GCF(12, 18, 30)

Factors of 12:

1, 2, 3, 4, 6, 12

Factors of 18:

1, 2, 3, 6, 9, 18

Factors of 30:

1, 2, 3, 5, 6, 10, 15, 30

Common factors:

1, 2, 3, 6

Therefore:

GCF(12, 18, 30) = 6

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Finding the LCM of Three Numbers

Find:

LCM(3, 4, 6)

Multiples of 3:

3, 6, 9, 12, 15, ...

Multiples of 4:

4, 8, 12, 16, ...

Multiples of 6:

6, 12, 18, ...

The smallest positive number appearing in all three lists is:

12

Therefore:

LCM(3, 4, 6) = 12

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Using Prime Factorization

Listing factors and multiples works well with smaller numbers.

For larger numbers, prime factorization is often more efficient.

Remember:

Prime factorization expresses a number as a product of prime factors.

For example:

24 = 2³ × 3

36 = 2² × 3²

These factorizations can be used to determine both the GCF and LCM.

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GCF Using Prime Factorization

Find:

GCF(24, 36)

Prime factorizations:

24 = 2³ × 3

36 = 2² × 3²

For the GCF, use only prime factors that appear in both numbers.

Both contain 2 and 3.

Use the smaller exponent of each shared prime.

For 2:

smaller exponent = 2

For 3:

smaller exponent = 1

Therefore:

GCF = 2² × 3

GCF = 4 × 3

GCF = 12

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Why Use the Smaller Exponent for GCF?

Consider:

24 = 2 × 2 × 2 × 3

36 = 2 × 2 × 3 × 3

The factors they share are:

2 × 2 × 3

There is no third 2 shared by both numbers.

There is no second 3 shared by both numbers.

Therefore:

GCF = 2 × 2 × 3 = 12

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LCM Using Prime Factorization

Now find:

LCM(24, 36)

Prime factorizations:

24 = 2³ × 3

36 = 2² × 3²

For the LCM, include enough prime factors to build either number.

Use the larger exponent of each prime.

For 2:

larger exponent = 3

For 3:

larger exponent = 2

Therefore:

LCM = 2³ × 3²

LCM = 8 × 9

LCM = 72

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Why Use the Larger Exponent for LCM?

The LCM must contain enough prime factors to be divisible by both original numbers.

24 requires:

2 × 2 × 2 × 3

36 requires:

2 × 2 × 3 × 3

To include everything needed by both:

2 × 2 × 2 × 3 × 3

Therefore:

LCM = 72

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Seeing GCF and LCM Together

For two numbers, prime factorization makes the relationship between GCF and LCM especially clear. For example, compare 24 and 36:

The GCF uses the prime factors shared by both numbers.

The LCM uses every prime factor needed to construct either number.

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A Prime Factorization Rule

For two or more numbers:

GCF: use shared primes with the smallest exponents.

LCM: use all required primes with the largest exponents.

This is one of the most useful rules for working with GCF and LCM.

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Worked Example 1: GCF

Find:

GCF(18, 30)

Prime factorizations:

18 = 2 × 3²

30 = 2 × 3 × 5

Shared prime factors:

2 × 3

Therefore:

GCF = 6

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Worked Example 2: LCM

Find:

LCM(18, 30)

Prime factorizations:

18 = 2 × 3²

30 = 2 × 3 × 5

Use the greatest required power of each prime:

2 × 3² × 5

Calculate:

2 × 9 × 5 = 90

Therefore:

LCM = 90

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Worked Example 3: GCF of Three Numbers

Find:

GCF(24, 36, 60)

Prime factorizations:

24 = 2³ × 3

36 = 2² × 3²

60 = 2² × 3 × 5

The shared primes are:

2 and 3

Use the smallest exponents:

2² × 3

Therefore:

GCF = 12

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Worked Example 4: LCM of Three Numbers

Find:

LCM(8, 12, 18)

Prime factorizations:

8 = 2³

12 = 2² × 3

18 = 2 × 3²

Use the largest exponent of each required prime:

2³ × 3²

= 8 × 9

= 72

Therefore:

LCM = 72

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Worked Example 5: One Number Is a Factor of the Other

Find the GCF and LCM of:

6 and 24

Since:

6 is a factor of 24

the greatest common factor is:

6

Since:

24 is already a multiple of 6

the least common multiple is:

24

Therefore:

GCF(6, 24) = 6

LCM(6, 24) = 24

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A Useful Pattern

If one number is a factor of another, such as:

8 and 40

then:

GCF = smaller number

and:

LCM = larger number

Therefore:

GCF(8, 40) = 8

LCM(8, 40) = 40

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When the GCF Is 1

Consider:

8 and 15

Factors of 8:

1, 2, 4, 8

Factors of 15:

1, 3, 5, 15

Their only common positive factor is:

1

Therefore:

GCF(8, 15) = 1

Numbers whose GCF is 1 are called coprime or relatively prime.

They do not need to be prime numbers themselves.

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Coprime Numbers

Consider:

8 and 9

8 is composite.

9 is composite.

But:

GCF(8, 9) = 1

Therefore:

8 and 9 are coprime.

This is an important distinction:

coprime does not mean both numbers are prime.

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GCF in Equal-Grouping Problems

Suppose a teacher has:

24 red markers

and:

36 blue markers

The teacher wants to make the greatest possible number of identical sets with no markers left over.

This is a GCF problem because the items are being divided into equal groups.

Find:

GCF(24, 36) = 12

Therefore:

12 identical sets

can be made.

Each set contains:

24 ÷ 12 = 2 red markers

and:

36 ÷ 12 = 3 blue markers

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How to Recognize a GCF Problem

Look for phrases such as:

• greatest number of equal groups
• largest equal pieces
• divide evenly
• identical groups
• no leftovers
• greatest possible size
• largest possible square tile

These often indicate a GCF problem.

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Real-World GCF Problem: Gift Bags

A store has:

42 pencils

and:

56 erasers

It wants to create the greatest possible number of identical gift bags with no items left over.

Find:

GCF(42, 56)

Prime factorizations:

42 = 2 × 3 × 7

56 = 2³ × 7

Shared factors:

2 × 7 = 14

Therefore:

14 gift bags

can be made.

Each bag contains:

42 ÷ 14 = 3 pencils

and:

56 ÷ 14 = 4 erasers

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Real-World GCF Problem: Cutting Ribbon

Two ribbons have lengths:

48 cm

and:

60 cm

They must be cut into equal pieces of the greatest possible length with no ribbon wasted.

Find:

GCF(48, 60)

Prime factorizations:

48 = 2⁴ × 3

60 = 2² × 3 × 5

GCF:

2² × 3 = 12

Therefore:

Each piece should be 12 cm long.

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Real-World GCF Problem: Square Tiles

A rectangular floor measures:

60 cm × 84 cm

You want to cover it using the largest possible identical square tiles without cutting any tiles.

The side length of each tile must divide both dimensions exactly.

Find:

GCF(60, 84)

Prime factorizations:

60 = 2² × 3 × 5

84 = 2² × 3 × 7

GCF:

2² × 3 = 12

Therefore:

The largest possible square tile is 12 cm × 12 cm.

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LCM in Repeating-Event Problems

Suppose one light flashes every:

4 seconds

and another flashes every:

6 seconds

They flash together now.

When will they next flash together?

This is an LCM problem because two repeating patterns must meet.

Find:

LCM(4, 6) = 12

Therefore:

They will flash together again after 12 seconds.

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How to Recognize an LCM Problem

Look for phrases such as:

• next occur together
• repeat together
• at the same time
• first time they meet again
• schedules
• repeating cycles
• every ___ minutes
• every ___ days

These often indicate an LCM problem.

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Real-World LCM Problem: Buses

Bus A arrives every:

12 minutes

Bus B arrives every:

18 minutes

Both arrive at the station at the same time.

When will they next arrive together?

Prime factorizations:

12 = 2² × 3

18 = 2 × 3²

LCM:

2² × 3² = 36

Therefore:

The buses will next arrive together in 36 minutes.

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Real-World LCM Problem: Exercise Schedule

One student goes swimming every:

4 days

and running every:

6 days

Both activities happen today.

When will both activities next occur on the same day?

Find:

LCM(4, 6) = 12

Therefore:

Both activities will occur together again in 12 days.

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Real-World LCM Problem: Machine Cycles

Machine A completes a cycle every:

15 seconds

Machine B completes a cycle every:

20 seconds

Machine C completes a cycle every:

30 seconds

When will all three finish a cycle together?

Prime factorizations:

15 = 3 × 5

20 = 2² × 5

30 = 2 × 3 × 5

LCM:

2² × 3 × 5

= 60

Therefore:

All three machines will finish together every 60 seconds.

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Choosing Between GCF and LCM

A major skill is deciding which one a problem requires.

Ask:

Am I dividing quantities into the largest possible equal groups or pieces?

Think:

GCF

Ask:

Am I looking for when repeating patterns will meet?

Think:

LCM

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GCF or LCM? Example 1

"You have 30 apples and 45 oranges. You want to make the greatest possible number of identical fruit baskets."

This asks for the greatest number of equal groups.

Use:

GCF

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GCF or LCM? Example 2

"One alarm sounds every 8 minutes and another every 12 minutes. When will they next sound together?"

This involves repeating cycles.

Use:

LCM

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GCF or LCM? Example 3

"Two boards measuring 72 cm and 96 cm must be cut into equal pieces of the greatest possible length."

This involves the largest equal pieces.

Use:

GCF

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GCF or LCM? Example 4

"Two satellites complete their orbits every 6 hours and 8 hours. If they are aligned now, when will they next be aligned at the starting point together?"

This involves repeating cycles.

Use:

LCM

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A Useful GCF–LCM Relationship

For two positive integers:

GCF × LCM = product of the two numbers

For example, consider:

12 and 18

GCF:

6

LCM:

36

Check:

6 × 36 = 216

and:

12 × 18 = 216

Therefore:

GCF × LCM = 12 × 18

This relationship can be used to check answers or find a missing GCF or LCM.

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Finding a Missing LCM

Suppose:

GCF(20, 30) = 10

Find the LCM.

Use:

GCF × LCM = 20 × 30

So:

10 × LCM = 600

Divide:

LCM = 60

Therefore:

LCM(20, 30) = 60

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Finding a Missing GCF

Suppose two numbers are:

18 and 24

and their LCM is:

72

Use:

GCF × 72 = 18 × 24

GCF × 72 = 432

Therefore:

GCF = 432 ÷ 72

GCF = 6

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Comparing the Three Main Methods

There are several useful ways to find GCF and LCM.

Listing factors or multiples

Best for:

• small numbers
• developing understanding
• checking answers

Prime factorization

Best for:

• larger numbers
• several numbers
• seeing mathematical structure
• working efficiently

Recognizing obvious relationships

Best when:

• one number is a factor of another
• numbers have simple familiar patterns

A strong mathematician chooses the method that fits the numbers.

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Worked Example 6

Find the GCF and LCM of:

16 and 24

Prime factorizations:

16 = 2⁴

24 = 2³ × 3

GCF uses smaller powers:

GCF = 2³ = 8

LCM uses larger powers:

LCM = 2⁴ × 3 = 48

Therefore:

GCF = 8

LCM = 48

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Worked Example 7

Find the GCF and LCM of:

45 and 60

Prime factorizations:

45 = 3² × 5

60 = 2² × 3 × 5

GCF:

3 × 5 = 15

LCM:

2² × 3² × 5

= 180

Therefore:

GCF = 15

LCM = 180

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Worked Example 8

Find the GCF and LCM of:

14 and 25

Prime factorizations:

14 = 2 × 7

25 = 5²

They share no prime factors.

Therefore:

GCF = 1

For the LCM, include all prime factors:

2 × 5² × 7

= 350

Therefore:

LCM = 350

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Worked Example 9

Find the GCF and LCM of:

30, 45, and 75

Prime factorizations:

30 = 2 × 3 × 5

45 = 3² × 5

75 = 3 × 5²

For the GCF, shared primes with smallest powers:

3 × 5 = 15

For the LCM, largest required powers:

2 × 3² × 5²

= 450

Therefore:

GCF = 15

LCM = 450

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Worked Example 10

Two bells ring every:

18 minutes

and:

24 minutes

They ring together at 9:00 a.m.

When will they next ring together?

Prime factorizations:

18 = 2 × 3²

24 = 2³ × 3

LCM:

2³ × 3² = 72

They will ring together again after:

72 minutes

72 minutes after 9:00 a.m. is:

10:12 a.m.

Therefore:

They next ring together at 10:12 a.m.

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Common Mistakes

Mistake 1: Choosing the greatest common multiple

Common multiples continue forever.

There is no greatest common multiple.

We want the:

least common multiple

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Mistake 2: Choosing the least common factor

1 is a common factor of every pair of positive integers.

That would usually not be useful.

We want the:

greatest common factor

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Mistake 3: Using the largest exponents for GCF

For GCF, use:

smallest shared exponents

For LCM, use:

largest required exponents

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Mistake 4: Including a prime in the GCF that is not shared

For:

12 = 2² × 3

20 = 2² × 5

the GCF is:

2² = 4

Do not include 3 or 5 because they are not shared.

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Mistake 5: Forgetting a prime when finding the LCM

The LCM must contain enough prime factors to be divisible by every original number.

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Mistake 6: Automatically using GCF because the word "greatest" appears

Read the meaning of the problem.

The wording may describe a quantity rather than tell you which operation to use.

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Error Analysis

A student finds:

GCF(24, 36)

using:

24 = 2³ × 3

36 = 2² × 3²

and writes:

GCF = 2³ × 3² = 72

This is incorrect.

The student used the largest exponents, which is the LCM method.

For the GCF, use the smallest shared exponents:

GCF = 2² × 3 = 12

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Another Error Analysis

A student is solving:

"Two lights flash every 6 seconds and 8 seconds. When will they next flash together?"

The student calculates:

GCF(6, 8) = 2

and answers:

2 seconds

This does not make sense because neither light even completes its first cycle in 2 seconds.

The problem involves repeating events meeting again.

Use the LCM:

LCM(6, 8) = 24

Therefore:

The lights flash together again after 24 seconds.

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Checking GCF Answers

A GCF must:

• divide every original number exactly
• be no larger than the smallest original number
• be the greatest factor satisfying both conditions

For:

GCF(30, 42) = 6

Check:

30 ÷ 6 = 5

42 ÷ 6 = 7

So 6 divides both numbers exactly.

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Checking LCM Answers

An LCM must:

• be divisible by every original number
• be at least as large as the greatest original number
• be the smallest positive multiple satisfying these conditions

For:

LCM(8, 12) = 24

Check:

24 ÷ 8 = 3

24 ÷ 12 = 2

So 24 is a multiple of both.

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A Reliable Problem-Solving Strategy

Step 1: Understand the situation.

Determine what the problem is asking.

Step 2: Decide whether it involves shared factors or shared multiples.

Equal groups or largest pieces usually suggest GCF.

Repeating events usually suggest LCM.

Step 3: Choose a method.

Use:

• factor lists
• multiple lists
• prime factorization

Step 4: Calculate.

Find the GCF or LCM.

Step 5: Check.

Make sure the result satisfies all original numbers.

Step 6: Interpret.

Explain what the answer means in the real-world situation.

Step 7: Include units where appropriate.

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Did You Know?

GCF and LCM are closely connected because both describe the multiplicative structure of numbers.

Prime factorization makes this especially clear.

The GCF asks:

What prime factors do these numbers share?

The LCM asks:

What prime factors are needed to build every number?

This is why prime factorization is such a powerful method for solving both types of problems.

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Key Terms

• Factor: Whole number that divides another number exactly.
• Multiple: Result of multiplying a number by a whole number.
• Common factor: Factor shared by two or more numbers.
• Greatest common factor (GCF): Largest positive factor shared by two or more numbers.
• Greatest common divisor (GCD): Another name for GCF.
• Highest common factor (HCF): Another name for GCF.
• Common multiple: Multiple shared by two or more numbers.
• Least common multiple (LCM): Smallest positive multiple shared by two or more numbers.
• Prime factorization: Expression of a number as a product of prime factors.
• Coprime: Two or more numbers whose GCF is 1.
• Shared factor: Factor found in all numbers being compared.
• Repeating cycle: Event occurring at regular intervals.

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Key Relationships

For prime factorizations:

GCF → shared primes with smallest exponents

LCM → all required primes with largest exponents

For two positive integers:

GCF × LCM = product of the two numbers

If one number is a factor of another:

GCF = smaller number

LCM = larger number

If two numbers share no prime factors:

GCF = 1

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Key Takeaways

• The GCF is the greatest positive factor shared by two or more numbers.
• The LCM is the least positive multiple shared by two or more numbers.
• GCF can be found by listing factors.
• LCM can be found by listing multiples.
• Prime factorization provides an efficient method for finding both GCF and LCM.
• For GCF, use prime factors shared by all numbers.
• For GCF, use the smallest exponent of each shared prime.
• For LCM, include every prime factor needed by any of the numbers.
• For LCM, use the largest exponent of each required prime.
• GCF problems often involve dividing quantities into equal groups.
• GCF problems can involve cutting materials into the largest equal pieces.
• LCM problems often involve repeating patterns, schedules, or cycles.
• Asking whether a problem involves grouping or repeating can help distinguish GCF from LCM.
• If one number is a factor of another, the smaller number is the GCF and the larger number is the LCM.
• Numbers with a GCF of 1 are called coprime or relatively prime.
• Coprime numbers do not have to be prime numbers themselves.
• The product of the GCF and LCM of two positive integers equals the product of the original numbers.
• GCF answers should divide all original numbers exactly.
• LCM answers should be divisible by all original numbers.
• Real-world GCF applications include packaging, grouping, cutting, tiling, and arranging.
• Real-world LCM applications include schedules, machine cycles, transportation, flashing lights, alarms, and repeating events.
• Prime factorization reveals why GCF and LCM methods work rather than simply providing rules to memorize.
• Understanding GCF and LCM provides an important foundation for simplifying fractions, finding common denominators, working with ratios, and solving algebraic problems.