Lesson 6.13Prime Factorization
Start hereWords, worked example, and sentence starters
Learning target I can write a number as a product of its prime factors using a factor tree.
Why This Matters The space station received 60 supply crates. Mission Control needs to break this quantity into its prime components so the sorting robots can distribute them into equally sized pods. Help the crew find the prime factorization of 60!
Start hereWords for this lesson
Read each word before you begin. Say it out loud.
| Word | What it means | Example |
|---|---|---|
| Prime FactorizationSpanish: Factorización prima | Writing a whole number as a product of only prime numbers. | 36 = 2 × 2 × 3 × 3 = 2² × 3² |
| Prime numberSpanish: Número primo | A number bigger than 1 that you can only divide by 1 and itself. | 7 has only two factors: 1 × 7. So 7 is prime. |
| Composite numberSpanish: Número compuesto | A number bigger than 1 that you can divide by more than just 1 and itself. | 12 = 1 × 12, 2 × 6, 3 × 4 — six factors, so 12 is composite |
| FactorSpanish: Factor | A factor of a whole number divides it evenly, with no remainder. For example, 3 is a factor of 12 because 12 ÷ 3 = 4. | Factors of 12: 1, 2, 3, 4, 6, 12 — each divides 12 evenly. |
| Factor treeSpanish: Árbol de factores | A picture that splits a number into its prime numbers, step by step. | 24 → 4 × 6 → (2 × 2) × (2 × 3) → 2 × 2 × 2 × 3 |
| ExponentSpanish: Exponente | A small number that tells how many times to multiply a number by itself. | 2³ means 2 × 2 × 2 = 8 |
How it worksWorked example
These numbers are not on your problems. The steps are. Follow them with your own numbers.
I want the prime factorization of 60. I start by splitting it: 60 = 6 × 10.
- 6 is not prime, so I break it: 6 = 2 × 3. Both 2 and 3 are prime, so I stop those branches.
- 10 is not prime, so I break it: 10 = 2 × 5. Both 2 and 5 are prime, so I stop.
- Now every factor is prime: 60 = 2 × 2 × 3 × 5.
- Using exponents, I write it as 2² × 3 × 5. I check: 2 × 2 × 3 × 5 = 60. Correct.
Now you trySame steps, your turn
The steps are the same as the worked example. The numbers are yours. Do the work.
Let's factor 12 together. What is one way to split 12 into two factors? (You might say 3 × 4.)
Say it and write itSentence starters
Finish each sentence out loud with a partner. Then use them in your writing.
- I know is prime because its only factors are and . I know is composite because it has factors like .
Word bank Prime FactorizationPrime numberComposite numberFactorFactor treeExponentprimecompositefactorsdivisibletwo factorsmore than two
Watch out!A common mistake
A common mistake in Prime Factorization is stopping the factor tree too soon
Lesson 6.13Prime Factorization
Level 0Most support · every hint printed
Learning target I can write a number as a product of its prime factors using a factor tree.
Why This Matters The space station received 60 supply crates. Mission Control needs to break this quantity into its prime components so the sorting robots can distribute them into equally sized pods. Help the crew find the prime factorization of 60!
How it worksWorked example
These numbers are not on your problems. The steps are. Follow them with your own numbers.
I want the prime factorization of 60. I start by splitting it: 60 = 6 × 10.
- 6 is not prime, so I break it: 6 = 2 × 3. Both 2 and 3 are prime, so I stop those branches.
- 10 is not prime, so I break it: 10 = 2 × 5. Both 2 and 5 are prime, so I stop.
- Now every factor is prime: 60 = 2 × 2 × 3 × 5.
- Using exponents, I write it as 2² × 3 × 5. I check: 2 × 2 × 3 × 5 = 60. Correct.
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1Write the name of the correct group on each line.
Sort each expression: is it a COMPLETE prime factorization, or does it still have composite factors?
- 2 × 2 × 3 (for 12)
- 2 × 3 × 5 (for 30)
- 2 × 2 × 2 × 3 (for 24)
- 4 × 3 (for 12)
- 6 × 5 (for 30)
- 8 × 3 (for 24)
Steps to try- 'Complete' means every single factor shown is prime. Even one composite factor makes the whole expression NOT complete.
- Scan each expression for numbers like 4, 6, or 8 — those are composite, so any expression containing them isn't complete.
- Take '4 × 3 (for 12)': 4 = 2 × 2, so 4 is composite. That card goes in NOT Complete. Check each other card's factors the same way.
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2Circle the letter of the best answer. Show how you know.
Which of the following is a prime number?
- A9
- B15
- C17
- D21
Steps to try- Re-read what 'prime' means: exactly two factors, 1 and itself. Any extra factor makes a number composite.
- Test each choice by dividing by small primes like 2, 3, 5, and 7. If anything divides in evenly, that number is composite.
- Start with 9: since 3 × 3 = 9, it has a factor besides 1 and 9, so 9 is composite. Cross it off and test the next choice the same way.
Show your work -
3Circle the letter of the best answer. Show how you know.
What is the prime factorization of 30?
- A2 × 3 × 5
- B2 × 15
- C3 × 10
- D5 × 6
Steps to try- A prime factorization uses ONLY prime numbers. Look at each choice: are ALL of its factors prime?
- Break 30 down step by step, and keep splitting any factor that is still composite until every factor is prime.
- Start with 30 = 2 × 15. Is 15 prime? No — so split 15 into two primes and collect everything you found.
Show your work -
4Circle the letter of the best answer. Show how you know.
What is the prime factorization of 18?
- A2 × 3 × 3
- B2 × 9
- C3 × 6
- D6 × 3
Steps to try- Check every factor in each choice — numbers like 9 and 6 are composite, so they can't appear in a prime factorization.
- Build a factor tree for 18: split it into two factors, then keep splitting anything composite until only primes remain.
- Begin with 18 = 2 × 9. The 9 isn't prime, so break 9 into two equal prime factors and list all the primes you end with.
Show your work
Explain your thinkingHow did you decide whether a number is prime or composite? What strategy did you use?
Sentence starter I know ___ is prime because its only factors are ___ and ___. I know ___ is composite because it has factors like ___.