What Is Prime Factorization?

Every whole number greater than 1 is either:

• a prime number, or
• a composite number.

Composite numbers can be broken into smaller factors. If we continue breaking those factors apart until every factor is prime, we have found the number's prime factorization.

For example:

12 = 2 × 2 × 3

Since 2 and 3 are both prime numbers, this is the prime factorization of 12.

Prime factorization reveals the basic multiplicative structure of a number.

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Review: What Is a Factor?

A factor is a whole number that divides another whole number exactly.

For example:

Factors of 12:

1, 2, 3, 4, 6, 12

because each divides 12 without leaving a remainder.

Factor pairs of 12 are:

1 × 12

2 × 6

3 × 4

Factors are important because prime factorization involves repeatedly breaking a number into factors.

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What Is a Prime Number?

A prime number is a whole number greater than 1 that has exactly two positive factors:

1 and itself

For example, the factors of 7 are:

1 and 7

Therefore:

7 is prime.

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Examples of Prime Numbers

The first several prime numbers are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Notice that the numbers become less regular as they increase.

There is no simple repeating pattern that generates all prime numbers.

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Is 2 Prime?

Yes.

The factors of 2 are:

1 and 2

Therefore, 2 has exactly two positive factors.

2 is prime.

It is also the only even prime number.

Every other even number greater than 2 is divisible by 2 and therefore has more than two factors.

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Is 1 Prime?

No.

The number 1 has only one positive factor:

1

A prime number must have exactly two positive factors.

Therefore:

1 is not prime.

It is also not composite.

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What Is a Composite Number?

A composite number is a whole number greater than 1 that has more than two positive factors.

For example, the factors of 12 are:

1, 2, 3, 4, 6, 12

Since 12 has more than two factors:

12 is composite.

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Prime, Composite, or Neither?

Every positive whole number can be classified.

1: Neither prime nor composite

2: Prime

3: Prime

4: Composite

5: Prime

6: Composite

7: Prime

8: Composite

9: Composite

10: Composite

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How Can We Test Whether a Number Is Prime?

To determine whether a number is prime, test whether smaller prime numbers divide it exactly.

Useful primes to test include:

2, 3, 5, 7, 11, ...

Divisibility rules can make this easier.

For example, consider:

51

Digit sum:

5 + 1 = 6

Since 6 is divisible by 3:

51 is divisible by 3

In fact:

51 = 3 × 17

Therefore:

51 is composite.

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Example: Is 29 Prime?

Test possible small prime factors.

29 is not even, so it is not divisible by 2.

Digit sum:

2 + 9 = 11

so it is not divisible by 3.

It does not end in 0 or 5, so it is not divisible by 5.

No smaller prime divides 29 exactly.

Therefore:

29 is prime.

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Prime Numbers as Building Blocks

Prime numbers are sometimes called the building blocks of whole numbers.

Why?

Because every composite number can be written as a product of prime numbers.

For example:

18 = 2 × 3 × 3

20 = 2 × 2 × 5

30 = 2 × 3 × 5

42 = 2 × 3 × 7

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What Is Prime Factorization?

The prime factorization of a number is an expression showing that number as a product of prime numbers.

For example:

24 = 2 × 2 × 2 × 3

Every factor on the right is prime.

Therefore:

2 × 2 × 2 × 3

is the prime factorization of 24.

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Factor Trees

A factor tree is a diagram used to organize prime factorization.

Start with the number you want to factor.

Break it into any pair of factors.

Then continue breaking composite factors apart until every branch ends with a prime number.

For example, 24 could begin as:

24 = 4 × 6

Then:

4 = 2 × 2

and:

6 = 2 × 3

So the prime factors are:

2, 2, 2, 3

Therefore:

24 = 2 × 2 × 2 × 3

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Exploring a Factor Tree

A factor tree can begin with different factor pairs and still reach the same prime factors.

For example, explore how 60 can be broken down completely into primes:

The important idea is that the final prime factors do not depend on which valid factor pair you choose first.

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Factor Tree Example: 36

Start:

36

One possible factor pair is:

36 = 6 × 6

Then:

6 = 2 × 3

and:

6 = 2 × 3

Therefore:

36 = 2 × 3 × 2 × 3

Rearrange:

36 = 2 × 2 × 3 × 3

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Different Factor Trees Give the Same Result

We could also start 36 using:

36 = 4 × 9

Then:

4 = 2 × 2

and:

9 = 3 × 3

Therefore:

36 = 2 × 2 × 3 × 3

The same prime factors appear.

This is an important property of prime factorization.

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How to Build a Factor Tree

Step 1: Write the number at the top.

Step 2: Choose any factor pair.

Step 3: Draw branches to the two factors.

Step 4: Circle or identify factors that are prime.

Step 5: Continue splitting composite factors.

Step 6: Stop when every branch ends in a prime number.

Step 7: Write all the prime factors as a multiplication expression.

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Worked Example 1: Prime Factorization of 18

Start with:

18

Choose:

18 = 2 × 9

2 is prime.

9 is composite:

9 = 3 × 3

Both 3s are prime.

Therefore:

18 = 2 × 3 × 3

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Worked Example 2: Prime Factorization of 28

Start:

28 = 4 × 7

7 is prime.

Break 4 apart:

4 = 2 × 2

Therefore:

28 = 2 × 2 × 7

Check:

2 × 2 × 7 = 28

Correct.

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Worked Example 3: Prime Factorization of 40

Start:

40 = 5 × 8

5 is prime.

Break 8:

8 = 2 × 4

Then:

4 = 2 × 2

Therefore:

40 = 2 × 2 × 2 × 5

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Worked Example 4: Prime Factorization of 45

Start:

45 = 5 × 9

5 is prime.

Break 9:

9 = 3 × 3

Therefore:

45 = 3 × 3 × 5

Check:

3 × 3 × 5 = 45

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Worked Example 5: Prime Factorization of 72

Start:

72 = 8 × 9

Break 8:

8 = 2 × 4

4 = 2 × 2

Break 9:

9 = 3 × 3

Therefore:

72 = 2 × 2 × 2 × 3 × 3

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Writing Prime Factorizations in Order

Prime factors are usually written from smallest to largest.

Instead of:

3 × 2 × 5 × 2

write:

2 × 2 × 3 × 5

For example:

60 = 2 × 2 × 3 × 5

This makes the factorization easier to read and compare.

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Using Exponents

Repeated prime factors can be written using exponents.

For example:

2 × 2 × 2 = 2³

and:

3 × 3 = 3²

Therefore:

72 = 2 × 2 × 2 × 3 × 3

can be written as:

72 = 2³ × 3²

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Understanding Exponential Form

Consider:

2⁴ × 3

This means:

2 × 2 × 2 × 2 × 3

Calculate:

2 × 2 × 2 × 2 = 16

Then:

16 × 3 = 48

Therefore:

48 = 2⁴ × 3

The exponent tells us how many times the prime factor is repeated.

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Prime Factorization of 100

One factor tree could begin:

100 = 10 × 10

Then:

10 = 2 × 5

for each branch.

Therefore:

100 = 2 × 5 × 2 × 5

Rearrange:

100 = 2 × 2 × 5 × 5

Using exponents:

100 = 2² × 5²

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Prime Factorization of 120

Start:

120 = 12 × 10

Factor 12:

12 = 3 × 4

4 = 2 × 2

Factor 10:

10 = 2 × 5

Therefore:

120 = 2 × 2 × 2 × 3 × 5

Using exponents:

120 = 2³ × 3 × 5

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Another Method: Repeated Division

Factor trees are not the only method for finding prime factorization.

We can repeatedly divide by the smallest possible prime.

Consider:

84

Divide by 2:

84 ÷ 2 = 42

Again:

42 ÷ 2 = 21

Now divide by 3:

21 ÷ 3 = 7

Finally:

7 ÷ 7 = 1

So the prime factors are:

2 × 2 × 3 × 7

Therefore:

84 = 2² × 3 × 7

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Choosing Prime Divisors

When using repeated division, useful primes to test include:

2, 3, 5, 7, 11, ...

Divisibility rules help identify which prime to try.

For example:

150

is even, so divide by 2:

150 ÷ 2 = 75

75 ends in 5, so divide by 5:

75 ÷ 5 = 15

Again:

15 ÷ 5 = 3

3 is prime.

Therefore:

150 = 2 × 3 × 5 × 5

or:

150 = 2 × 3 × 5²

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Checking a Prime Factorization

Always multiply the prime factors back together.

Suppose:

90 = 2 × 3² × 5

Check:

3² = 9

Then:

2 × 9 × 5 = 90

Therefore, the factorization is correct.

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The Unique Prime Factorization Idea

Consider:

60

We could begin:

60 = 6 × 10

or:

60 = 4 × 15

or:

60 = 3 × 20

Different factor trees are possible.

But when every branch is broken into prime factors, each method gives:

60 = 2 × 2 × 3 × 5

The order might differ, but the prime factors are the same.

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Fundamental Theorem of Arithmetic

A major mathematical idea says:

Every whole number greater than 1 can be expressed as a product of prime numbers in essentially one unique way.

"Essentially" means that the order does not matter.

For example:

2 × 3 × 5

and:

5 × 2 × 3

represent the same prime factorization.

This result is called the Fundamental Theorem of Arithmetic.

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Why Is Prime Factorization Useful?

Prime factorization helps reveal relationships between numbers.

It can be used to:

• identify factors
• find common factors
• find the greatest common factor
• find common multiples
• find the least common multiple
• simplify fractions
• work with roots
• solve number problems
• study patterns in mathematics

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Using Prime Factorization to Find the GCF

Consider:

24 and 36

Prime factorization of 24:

24 = 2³ × 3

Prime factorization of 36:

36 = 2² × 3²

The prime factors they share are:

2² × 3

Therefore:

GCF = 12

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Why the GCF Method Works

Write:

24 = 2 × 2 × 2 × 3

and:

36 = 2 × 2 × 3 × 3

Both numbers contain:

2 × 2 × 3

Therefore:

2 × 2 × 3 = 12

is a common factor.

It is the largest combination of prime factors shared by both numbers.

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Using Prime Factorization to Find the LCM

Consider:

12 and 18

Prime factorizations:

12 = 2² × 3

18 = 2 × 3²

For the LCM, include enough prime factors to contain both factorizations:

2² × 3²

Calculate:

4 × 9 = 36

Therefore:

LCM(12, 18) = 36

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Prime Factorization and Fractions

Prime factorization can help simplify fractions.

Consider:

18/24

Prime factorizations:

18 = 2 × 3 × 3

24 = 2 × 2 × 2 × 3

Common prime factors include:

2 × 3 = 6

Divide numerator and denominator by 6:

18/24 = 3/4

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Prime Factorization and Square Roots

Prime factorization can also help simplify square roots.

Consider:

√36

Prime factorization:

36 = 2² × 3²

Therefore:

√36 = √(2² × 3²)

= 2 × 3

= 6

Prime factors reveal the perfect-square structure of the number.

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Recognizing Perfect Squares

Prime factorization can help identify perfect squares.

Consider:

144 = 2⁴ × 3²

Every exponent is even.

Therefore, 144 is a perfect square.

In fact:

144 = 12²

This idea becomes useful in algebra and work with radicals.

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Real-World Problem 1: Equal Groups

A teacher has:

48 pencils

and:

60 pens

and wants to create the greatest possible number of identical sets with no items left over.

Prime factorizations:

48 = 2⁴ × 3

60 = 2² × 3 × 5

Common prime factors:

2² × 3 = 12

Therefore:

12 identical sets

can be created.

Each set contains:

48 ÷ 12 = 4 pencils

and:

60 ÷ 12 = 5 pens

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Real-World Problem 2: Repeating Events

One machine completes a cycle every:

12 minutes

Another completes a cycle every:

18 minutes

Prime factorizations:

12 = 2² × 3

18 = 2 × 3²

LCM:

2² × 3² = 36

Therefore:

The machines will complete a cycle together every 36 minutes.

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Real-World Problem 3: Arranging Tiles

Suppose:

72 tiles

must be divided into equal groups.

Prime factorization:

72 = 2³ × 3²

This tells us that every factor of 72 can be built from combinations of:

2 × 2 × 2 × 3 × 3

This information can be used to determine possible equal arrangements.

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Worked Example 6

Find the prime factorization of:

54

Start:

54 = 6 × 9

Then:

6 = 2 × 3

and:

9 = 3 × 3

Therefore:

54 = 2 × 3 × 3 × 3

Using exponents:

54 = 2 × 3³

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Worked Example 7

Find the prime factorization of:

96

Repeatedly divide by 2:

96 ÷ 2 = 48

48 ÷ 2 = 24

24 ÷ 2 = 12

12 ÷ 2 = 6

6 ÷ 2 = 3

3 is prime.

Therefore:

96 = 2 × 2 × 2 × 2 × 2 × 3

or:

96 = 2⁵ × 3

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Worked Example 8

Find the prime factorization of:

225

Since it ends in 5:

225 = 5 × 45

Then:

45 = 5 × 9

and:

9 = 3 × 3

Therefore:

225 = 3 × 3 × 5 × 5

or:

225 = 3² × 5²

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Worked Example 9

Is 91 prime or composite?

Test small prime factors.

Not divisible by 2.

Digit sum:

9 + 1 = 10

so not divisible by 3.

Does not end in 0 or 5.

Try 7:

91 ÷ 7 = 13

Therefore:

91 = 7 × 13

So:

91 is composite.

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Worked Example 10

Find the prime factorization of:

210

We can write:

210 = 21 × 10

Then:

21 = 3 × 7

and:

10 = 2 × 5

Therefore:

210 = 2 × 3 × 5 × 7

All four factors are prime.

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Common Mistakes

Mistake 1: Stopping before every factor is prime

For example:

24 = 4 × 6

is a factorization, but not a prime factorization.

Both 4 and 6 are composite.

Continue:

24 = 2 × 2 × 2 × 3

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Mistake 2: Including 1 in the prime factorization

Incorrect:

12 = 1 × 2 × 2 × 3

Although mathematically the product is 12, 1 is not prime.

Prime factorization uses only prime factors.

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Mistake 3: Thinking 1 is prime

1 has only one positive factor.

Therefore:

1 is neither prime nor composite.

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Mistake 4: Thinking every odd number is prime

For example:

9 = 3 × 3

15 = 3 × 5

21 = 3 × 7

These are odd but composite.

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Mistake 5: Thinking every even number is composite

Almost true, but:

2

is even and prime.

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Mistake 6: Forgetting repeated factors

Incorrect:

24 = 2 × 3

Correct:

24 = 2 × 2 × 2 × 3

Each occurrence of a prime factor must be included.

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Error Analysis

A student writes:

36 = 6 × 6

and says this is the prime factorization.

The multiplication is correct, but 6 is not prime.

Continue factoring:

6 = 2 × 3

for both factors.

Therefore:

36 = 2 × 3 × 2 × 3

Rearrange:

36 = 2² × 3²

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Another Error Analysis

A student writes:

48 = 2³ × 3

Check:

2³ × 3 = 8 × 3 = 24

This does not equal 48.

The correct prime factorization is:

48 = 2⁴ × 3

because:

2⁴ × 3 = 16 × 3 = 48

Multiplying the prime factors back together is an effective way to check your work.

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A Reliable Prime Factorization Strategy

Step 1: Decide whether the number is already prime.

Step 2: If it is composite, choose a factor pair.

Step 3: Break every composite factor into smaller factors.

Step 4: Continue until every factor is prime.

Step 5: List the prime factors from smallest to largest.

Step 6: Use exponents for repeated prime factors if appropriate.

Step 7: Multiply the factors to check your answer.

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Choosing an Efficient Factor Pair

Any correct factor pair works, but some choices make the process faster.

For example, to factor:

72

we could choose:

72 = 8 × 9

This is convenient because:

8 = 2 × 2 × 2

and:

9 = 3 × 3

So:

72 = 2³ × 3²

Recognizing familiar factors can make factor trees much faster.

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Using Divisibility Rules

Divisibility rules can help choose factors efficiently.

If a number is even:

try 2.

If its digit sum is divisible by 3:

try 3.

If it ends in 0 or 5:

try 5.

For example:

330

is even:

330 = 2 × 165

165 ends in 5:

165 = 5 × 33

Then:

33 = 3 × 11

Therefore:

330 = 2 × 3 × 5 × 11

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Did You Know?

Prime numbers continue forever.

There is no largest prime number.

Mathematicians have known this for more than 2,000 years.

Prime numbers also play an important role in modern computing.

Some encryption systems use mathematical problems involving very large prime numbers and their products. Multiplying large primes is straightforward for computers, while reversing certain large-number problems can be much more difficult.

This makes prime numbers useful far beyond classroom mathematics.

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Key Terms

• Factor: Whole number that divides another whole number exactly.
• Prime number: Whole number greater than 1 with exactly two positive factors.
• Composite number: Whole number greater than 1 with more than two positive factors.
• Prime factor: Factor that is also a prime number.
• Prime factorization: Expression of a whole number as a product of prime numbers.
• Factor pair: Two factors whose product equals a given number.
• Factor tree: Diagram showing repeated factorization until only prime factors remain.
• Exponent: Number showing how many times a factor is multiplied by itself.
• Product: Result of multiplication.
• Divisibility: Whether one number divides another exactly.
• Greatest common factor (GCF): Greatest factor shared by two or more numbers.
• Least common multiple (LCM): Smallest positive multiple shared by two or more numbers.
• Fundamental Theorem of Arithmetic: Principle that every integer greater than 1 has a unique prime factorization apart from the order of its factors.

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Key Relationships

Prime number:

exactly two positive factors

Composite number:

more than two positive factors

Prime factorization example:

24 = 2 × 2 × 2 × 3

Exponential form:

24 = 2³ × 3

Another example:

180 = 2² × 3² × 5

To check:

4 × 9 × 5 = 180

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Key Takeaways

• Prime numbers have exactly two positive factors: 1 and themselves.
• Composite numbers have more than two positive factors.
• The number 1 is neither prime nor composite.
• The number 2 is the only even prime number.
• Not every odd number is prime.
• Prime numbers are the basic multiplicative building blocks of whole numbers.
• Prime factorization expresses a number as a product containing only prime factors.
• Factor trees provide a visual way to organize prime factorization.
• A factor tree is complete only when every final branch is prime.
• Different factor trees for the same number produce the same collection of prime factors.
• Prime factors are usually written from smallest to largest.
• Repeated prime factors can be written efficiently using exponents.
• Divisibility rules can help identify useful factors quickly.
• Repeated division provides another method for finding prime factorization.
• Multiplying the prime factors back together is an effective way to check an answer.
• Every whole number greater than 1 has a unique prime factorization apart from the order of its factors.
• Prime factorization can be used to find common factors and the GCF.
• Prime factorization can be used to find common multiples and the LCM.
• Prime factorization can help simplify fractions.
• Prime factorization can help identify and simplify perfect squares and roots.
• Prime factorization is useful in number theory, algebra, computing, and many other areas of mathematics.
• Understanding prime factorization provides an important foundation for later work with fractions, ratios, roots, exponents, algebra, and divisibility.