What is Prime Factorization? | Math with Mr. J

Welcome to Math with Mr. J. In this video, I'm going to cover what prime factorization is, and we will go through a few examples. 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 1, 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 are 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 times 15 equals 30. So let's break this down into 2 and 15. So 2 and 15. Now you may be thinking, 5 times 6 equals 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 2 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 2 is a prime number. The only factors are 1 and 2. So 2 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 2 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 2 factors. Well, 3 and 5 are factors of 15. 3 times 5 is 15. So 3 and 5. Both 3 and 5 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 times 3 times 5. If we have the correct prime numbers, they will multiply to equal our original number. 2 times 3 is 6. Times 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 2, 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 5 and 6 and see where we end up. So we have 30. And let's start with the factors 5 and 6. 5 is prime, so we are done there. And then 6 we can break down to 2 and 3. 2 is prime. And 3 is prime. So we are done. And we get the same prime factors. 2, 3, and 5. So we get the same thing that way as well. Let's move on to number 2 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 times 7 equals 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 times 3 equals 9. 3 is a factor of 9. So we have 3 and 3. 3 is prime, so we are done here. Nothing else can be broken down. We have all prime numbers. So 63 equals 3 times 3 times 7. And let's double check this. Let's multiply. 3 times 3 is 9 times 7 is 63. So our prime factorization is correct. Lastly, let's move on to number 3 where we have 48. Now 48 is an even number. And whenever we have an even number, we have a factor of 2 that we can start with. So we can start with 2 and 24 here. 2 times 24 equals 48. Now do we have to start with a 2 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 2 for number 1, let's try something else here for number 3. So what other factors of 48 can we start with? Well, 4 and 12 are factors of 48. 4 times 12 equals 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 times 2 is 4. So 2 is a factor. So we have 2 and 2. So 2 is prime. So we are done over here. Now for 12. Let's think of two factors of 12. How about 3 and 4? 3 times 4 is 12. So 3 and 4. 3 is prime. So we are done here. But 4 is composite. So let's break 4 down into 2 and 2. 2 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 equals 2 times 2 times 2 times 2. We have 4 twos times 3. And let's double check here. 2 times 2 is 4 times 2 is 8 times 2 is 16 times 3 is 48. So we are correct. So there you have it. There's what prime factorization is. I hope that helped. Thanks so much for watching. Until next time. Peace.