Prime Factorization (Practice Problems Included) | Math with Mr. J

[Music] Welcome to Math with Mr. J. In this video, I'm going to cover what prime factorization is. We will go through some examples together and then we will end with some practice problems. Now, when it comes to the prime factorization of a number, we break the number down into its prime factors. These are the prime numbers that multiply to equal the original number. So, we are decomposing a number, breaking a number down to only prime numbers. This will make a lot more sense as we go through our examples. And remember, prime numbers only have two factors. 1 and the number itself. For example, 2, 3, 5, 7, and 11 are all prime numbers. Let's jump into our examples. Starting with number one where we have 30. Now 30 is a composite number meaning 30 has factors other than 1 and 30. So we can break this down into other factors. And we keep breaking this down. We keep going until we have all prime numbers. The method we are going to use here is called a factor tree. Now to start this off we need two factors of 30. So, we're looking for a factor pair. We need two numbers that multiply to equal 30. We will branch off of 30 with those. Then, we will continue to work our way down until we only have prime numbers. 2 and 15 are factors of 30. 2 * 15 = 30. So, let's break this down into two and 15. So 2 and 15. Now you may be thinking 5 * 6 = 30 as well. Could we have done that? Absolutely. That works too. Depending on the number we are working with, there can be different paths we can take. And it does not matter if two is on the right and 15 is on the left. We will get the same prime numbers either way. And that's what's important here. Now two is a prime number. The only factors are 1 and two. So two cannot be broken down anymore. We are done over here. We are done with that branch off of 30. So I'm going to circle two to show that we have a prime number there. 15 on the other hand is a composite number. We can continue to break that down. So let's think of two factors. Well, 3 and 5 are factors of 15. 3 * 5 is 15. So 3 and 5. Both three and five are prime numbers. So we are done. We cannot break this down any further. We are done with the factor tree. This is the prime factorization of 30. So we can write this out as 30 equals and then I'll put our prime factors in order from least to greatest and we are multiplying them. So we have 2 * 3 * 5. If we have the correct prime numbers they will multiply to equal our original number. 2 * 3 is 6 * 5 is 30. So we are good to go here. We are correct. So this is the prime factorization of 30. Now before moving on to number two, let's go through this a different way in order to see if we end up with the same prime factors. Let's try this by starting with five and six and see where we end up. So we have 30 and let's start with the factors five and six. Five is prime. So we are done there. And then six we can break down to two and three. Two is prime and three is prime. So we are done and we get the same prime factors 2 3 and five. So we get the same thing that way as well. Let's move on to number two where we have 63. 63 is a composite number. So, we can break this down. Think of two factors, two numbers that will multiply to equal 63. And we'll start there. Well, 9 and 7 are factors. 9 * 7 equ= 63. So, 9 and 7. Now, 9 is composite. So, we can continue to break that down. But 7 is prime. So, we are done here. Now going back to 9. 3 * 3 = 9. 3 is a factor of nine. So we have three and three. Three is prime. So we are done here. Nothing else can be broken down. We have all prime numbers. So 63 equ= 3 * 3 * 7. And let's double check this. Let's multiply. 3 * 3 is 9 * 7 is 63. So our prime factorization is correct. Lastly, let's move on to number three where we have 48. Now 48 is an even number and whenever we have an even number we have a factor of two that we can start with. So we can start with 2 and 24 here. 2 * 24 = 48. Now do we have to start with a two if we have an even number? No. But it can be a good starting point. So something to keep in mind since we started with two for number one, let's try something else here for number three. So what other factors of 48 can we start with? Well, 4 and 12 are factors of 48. 4 * 12 = 48. So let's start with 4 and 12. So 4 and 12. 4 and 12 are both composite. So, we need to break both down. Let's start with 4. Well, 2 * 2 is 4. So, two is a factor. So, we have 2 and 2. So, 2 is prime. So, we are done over here. Now, 4 12. Let's think of two factors of 12. How about 3 and 4? 3 * 4 is 12. So three and four. Three is prime. So we are done here. But four is composite. So let's break four down into two and two. Two is prime. So we are done here and here. We cannot break this down any further. So this is the prime factorization of 48. Let's write this out as 48 equ= 2 * 2 * 2 * 2 we have four 2's * 3. And let's double check here. 2 * 2 is 4 * 2 is 8 * 2 is 16 * 3 is 48. So we are correct. So, now that we've gone through some examples together, it's time for you to try some practice problems on your own. So, have paper and a pencil ready. Here are your practice problems. There are three numbers. Find the prime factorization of each. I'll give you 2 minutes and then we will go over the answers. Go ahead and start. [Music] Happy birthday. Hey, hey, hey. [Music] [Music] Okay, so that's 2 minutes. Let's go over the answers. So, here are our answers. Remember, there are different paths we can take depending on the number. So, if you have something different than what I wrote out, that's okay as long as you have the same prime numbers. Taking a look at number one, we have 36. 36 has a handful of different factors we can start with. 2 and 18, 3 and 12, 4 and 9, or 6 and 6. I have factor trees for 6 and 6 and 4 and 9 written out. But if you used any of the other options, that's fine as long as you have 2 * 2 * 3 * 3. Moving on to number two, we have 27. The factors we can use to break this down are 3 and 9 and we end up with 3 * 3 * 3. Lastly, for number three, we have 60. 60 has a handful of factors. We can start with as well 2 and 30, 3 and 20, 4 and 15, 5 and 12 or 6 and 10. I have factor trees for 2 and 30 and 6 and 10 written out. No matter how you did it, you should have 2 * 2 * 3 * 5. So there you have it. There's what prime factorization is. I hope that helped. Thanks so much for watching. Until next time, peace.