Prime Factorization Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Prime Factorization.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Writing a whole number as a product of prime numbers; every composite number has exactly one such representation (up to order).

Break a number into building blocks that cannot be split further (primes).

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Prime factorization breaks a whole number into the prime building blocks that multiply back to it, in one and only one way.

Common stuck point: The procedure for prime factorization is the easy part; the trap is stopping before all factors are prime. Asking "Is every factor in my answer a prime number that can't be broken down any further?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is every factor in my answer a prime number that can't be broken down any further?

Worked Examples

Example 1

easy

Find the prime factorization of

360

using a factor tree.

Answer

360 = 2^3 \times 3^2 \times 5

First step

Start:

360 = 2 \times 180

Full solution

2
$180 = 2 \times 90$ ; $90 = 2 \times 45$ ; $45 = 3 \times 15$ ; $15 = 3 \times 5$ .
3
Collect all prime factors: $2 \times 2 \times 2 \times 3 \times 3 \times 5 = 2^3 \times 3^2 \times 5$ .
4
Verify: $8 \times 9 \times 5 = 72 \times 5 = 360$ . ✓

Prime factorization expresses any composite number as a product of prime numbers. The result is unique by the Fundamental Theorem of Arithmetic — no matter what order you factor, the prime factors and their exponents are always the same.

Example 2

hard

Use prime factorization to find

\gcd(180, 252)

and

\text{lcm}(180, 252)

Example 3

medium

Use prime factorization to find

\gcd(60, 84)

Example 4

medium

Use prime factorization to find

\text{lcm}(60, 84)

Example 5

medium

How many positive divisors does

72

have?

Example 6

medium

Find all divisors of

50

using its prime factorization.

Example 7

hard

Find the prime factorization of

1000

Example 8

hard

Use prime factorization to find

\gcd(120, 200)

and

\text{lcm}(120, 200)

Example 9

hard

How many positive divisors does

360

have?

Example 10

hard

Use prime factorization to determine if

\sqrt{500}

is rational.

Example 11

challenge

What is the smallest positive integer divisible by every integer from

1

through

10

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Write the prime factorization of

84

and use it to find all of its factors.

Example 2

medium

A rectangular garden can be arranged as a whole-number rectangle in exactly

6

different ways (including

1 \times n

and

n \times 1

as different). What is the smallest possible area for this garden?

Example 3

easy

7

prime?

Example 4

easy

Find the prime factorization of

12

Example 5

easy

Find the prime factorization of

18

Example 6

easy

List all factors of

12

Example 7

easy

15

prime or composite?

Example 8

easy

Find the prime factorization of

20

Example 9

easy

List the first five prime numbers.

Example 10

easy

Find the prime factorization of

30

Example 11

medium

Find the prime factorization of

72

Example 12

medium

Find the GCD of

24

and

36

using prime factorization.

Example 13

medium

Find the LCM of

8

and

12

using prime factorization.

Example 14

medium

Find the prime factorization of

100

Example 15

medium

How many divisors does

36

have?

Example 16

medium

51

prime? Show why or why not.

Example 17

medium

Find the prime factorization of

84

Example 18

medium

Simplify

\sqrt{72}

using prime factorization.

Example 19

medium

Find the prime factorization of

1000

Example 20

challenge

What is the smallest number with exactly

6

divisors?

Example 21

challenge

2^{10} \times 3^4

a perfect square? A perfect cube?

Example 22

challenge

What is the smallest positive integer to multiply

90

by to make a perfect square?

Example 23

easy

Find the prime factorization of

24

Example 24

easy

Find the prime factorization of

45

Example 25

easy

Find the prime factorization of

63

Example 26

easy

Find the prime factorization of

100

Example 27

easy

Find the prime factorization of

48

Example 28

medium

Find the prime factorization of

144

Example 29

medium

Find the prime factorization of

210

Example 30

medium

What is the prime factorization of

99

Example 31

medium

What is the prime factorization of

128

Example 32

hard

Find the prime factorization of

675

Example 33

hard

Find the prime factorization of

2310

Example 34

hard

True or false: every integer greater than

1

has a unique prime factorization. Name the theorem.

Example 35

easy

Find the prime factorization of

56

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Related Concepts

Prime Numbers Composite Numbers Divisibility Intuition

Background Knowledge

These ideas may be useful before you work through the harder examples.

prime numberscomposite numbersdivisibility intuition