Neft Teacher · Learn It

Unit 6 · Standard 6.NOS.4

Prime Factorization

Lesson 6-13 · Learn It — how the math works

Why we're learning this

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!

✱ Math Workbench

Learn It

What is prime factorization?

Prime factorization means writing a number as a product of prime numbers only. A prime number can be divided only by 1 and itself, like 2, 3, 5, and 7.

Picture itSee what the math looks like before you read about it

Engineers on Station Helios need to break 84 replacement circuit boards into equal shipping crates, and the crates must be built using only prime-numbered groupings.
Where you see this in real lifeEngineers on Station Helios need to break 84 replacement circuit boards into equal shipping crates, and the crates must be built using only prime-numbered groupings.
Prime factors of 60
2235
60 = 2 × 2 × 3 × 5 = 2² × 3 × 5

Watch me solve itI do — read one step at a time

  1. I want the of 60. I start by splitting it:60 = 6 × 10
    Factor tree for 60. 60 splits into 6 and 10. not prime yet prime — stop here 60 6 10Keep splitting every circle that is not prime.
    The factor tree so far
  2. 6 is not , so I break it: 6 = 2 × 3. Both 2 and 3 are prime, so I stop those branches.6 = 2 × 3
    Factor tree for 60. 60 splits into 6 and 10; 6 splits into 2 and 3. not prime yet prime — stop here 60 6 10 2 3Keep splitting every circle that is not prime.
    The factor tree so far
  3. 10 is not , so I break it: 10 = 2 × 5. Both 2 and 5 are prime, so I stop.10 = 2 × 5
    Factor tree for 60. 60 splits into 6 and 10; 6 splits into 2 and 3; 10 splits into 2 and 5. Every branch ends in a prime: 60 = 2 × 2 × 3 × 5. not prime yet prime — stop here 60 6 10 2 3 2 5 60 = 2 × 2 × 3 × 5
    The factor tree so far
  4. Now every is :60 = 2 × 2 × 3 × 5
  5. Using , I write it as 2² × 3 × 5. I check: 2 × 2 × 3 × 5 = 60. Correct.2 × 2 × 3 × 5 = 60

Watch out — mistakes students make here

  • 20 still breaks down — a prime factorization stops only at primes.
  • 4 and 10 aren't prime — keep splitting until every factor is prime.
  • 8 isn't prime — break it down further into primes.

Try it with meWe do — you answer first, then check

Same steps, new numbers.

  1. Let's 12 together. What is one way to split 12 into two factors?
    Check this step
    (You might say 3 × 4.)
  2. Is 3 ?
    Check this step
    Yes, so we keep it. Is 4 ? No — what is 4 split apart? (4 = 2 × 2.)
  3. So put the together: 12 = 2 × 2 × 3. Every factor is prime, so we are done.12 = 2 × 2 × 3

On your ownYou do — solve this one by yourself

Before you start, remember:

  • Remember: keep splitting until every branch is a .

Solve: Two students found different for 60. Student A started with 2 × 30. Student B started with 6 × 10. Which statement is true?

My answer

Stuck? Open one hint at a time.

Hint 1

Notice the question isn't asking which START is right — it asks what happens when both finished trees are compared.

Hint 2

Finish BOTH trees yourself: keep splitting every branch down to primes, then compare the two final lists of prime factors.

Hint 3

Student A: 60 = 2 × 30, then 30 = 2 × 15, then 15 = 3 × 5. Now finish Student B's tree from 6 × 10 and compare the prime lists.

Check my answer
✅ Answer: Both get the same prime factorization: 2 × 2 × 3 × 5
The Fundamental Theorem of Arithmetic says every composite number has exactly one prime factorization. No matter how you start the factor tree, you always end with 2 × 2 × 3 × 5.

Turn & TalkSay it out loud with a partner

Mission Control needs to break the 60 supply crates into prime parts. Is 60 a or a , and how can you tell before you even start the ?

Try starting with:

  • 60 is a ___ number because it has ___.60 es un número ___ porque tiene ___.
  • I can tell before factoring because ___.Puedo saberlo antes de factorizar porque ___.
Use these wordsprime numbercomposite numberfactordivisiblefactor tree