Step by step
Pick two numbers that multiply to 360
Start the tree with any factor pair. Mr DJ notices 360 is even, so 2 would work, but he also sees that 360 is divisible by 10. He picks 10 × 36.
▶ Watch this step 0:00Check each branch: prime or composite?
10 is composite, so it isn't a prime number, and 36 is composite too. Any composite number has to keep breaking down. 10 = 2 × 5, and both 2 and 5 are prime, so those branches stop.
▶ Watch this step 0:22Break down 36 any way you like
For 36, Mr DJ says you could use 18 × 2, 9 × 4 or 6 × 6, and you'll get the same answer no matter what. He picks 6 × 6. Each 6 is still composite, so each one becomes 2 × 3.
▶ Watch this step 0:46Circle the primes where the tree stops
Circle every 2, 3 and 5 at the end of a branch. Mr DJ likes circling the primes because when he goes back to write the prime factorization, he can clearly see which numbers he's working with.
▶ Watch this step 1:08Write out every circled prime
Collect all the circled numbers from the tree: 2 × 2 × 2 × 3 × 3 × 5. That's the prime factorization of 360. Multiplying them back together gives 360.
▶ Watch this step 1:29Rewrite it with exponents
Three 2s make 2³, and two 3s make 3². There is one 5, which you can write as 5 or 5¹; either one is the same thing. So 360 = 2³ × 3² × 5.
▶ Watch this step 1:50
“You'll get the same answer no matter what.”
Worked example · with Digit the Fox
Try one more, step by step
Problem: Find the prime factorization of 240 using a factor tree.
- 240 is divisible by 10, so start with 240 = 10 × 24.
- 10 is composite: 10 = 2 × 5. Circle the 2 and the 5.
- 24 is composite: 24 = 4 × 6.
- 4 is composite: 4 = 2 × 2. Circle both 2s.
- 6 is composite: 6 = 2 × 3. Circle the 2 and the 3.
- Collect the circled primes: 2 × 2 × 2 × 2 × 3 × 5.
- Four 2s make 2⁴, so write 2⁴ × 3 × 5.
Answer: 240 = 2⁴ × 3 × 5
Check: 2⁴ = 16, and 16 × 3 × 5 = 240.
Common mistakes
- Thinking there is one right pair to start with, or that a different split changes the answer. Any factor pair works. Start with whichever pair you notice first; the primes at the bottom of the tree come out the same. Mr DJ warns about this
- Stopping a branch at a composite number, like leaving a 6 at the bottom. If a number is composite, break it down again: 6 = 2 × 3. Mr DJ warns about this
- Dropping a prime when you write the answer, like writing 2² for 360. Circle each prime as its branch stops, then count: 360 has three 2s, two 3s and one 5.
- Reading an exponent as multiplication, like thinking 2³ is 2 × 3 = 6. 2³ means 2 × 2 × 2 = 8. The exponent counts how many times the prime appears.
Questions
Does it matter which two factors you start a factor tree with?
No. Any pair of factors works, and the primes at the bottom of the tree come out the same. Mr DJ could have started 360 with 2, since it's even, but chose 10 × 36.
What is the prime factorization of 360?
360 = 2 × 2 × 2 × 3 × 3 × 5. With exponents, that's 2³ × 3² × 5, where the 5 can also be written 5¹.
Read the full transcript
0:00 Welcome to Math Academy. So, 360. Okay. And when now we want to pick two numbers that go into 360. Now, we could do, you know, two uh uh again because we know 360 is a uh even number. Um but I'm also recognizing right now that um uh 360 is divisible by 10 as well. So, let's do that. Let's do 10 * 36. Okay, now 10 is composite, so it's not a prime number. And 36 is also composite, so it's not a prime number. So we have to keep breaking both of those down. All right. And 2 × 5 gets
0:46 us 10. Okay. And now 2 is prime. 5 is prime. And then for six, we can do um you know, you can do uh 18 × 2 or you can do uh 9 × 4 or you can do 6 × 6. Um it's up to you. You'll get the same answer no matter what. Um 6 × 6. Great. And uh 6 is not prime. It's composite. So we got to keep going with both sixes. And we'll get 2 × 3 and 2 × 3. And let's uh circle both of those to be consistent here. All right. I like circling the prime numbers when the tree stops because when I go back and I'm doing the prime factorization part, it's it's really helpful um to like really see
1:31 what what numbers I'm working with here. All right. So, um so let's go in here. So the prime factorization of 360 is going to be uh 2 * 2 * 2 * 3 * 3 * 5. And if we put it all together, we're going to get see the uh three of them. That's 2 3R* 3 2 × 5 or you can write 5 to the 1st. Either one is the same thing. Um so the prime factorization of 360 is 2
2:17 3R × 3 2 × 5 1st. All right.
From the video’s captions.
