Video: Finding the Prime Factorization of a Number

With a series of examples using factor trees and the method of dividing by prime numbers, we carefully walk you through the process of finding the prime factorization of a composite number then leave you with a few tips at the end.


Video Transcript

Let’s look at finding the prime factorization of a number. Before we get started, there are two words we need to know here. And that would be “factor” and “prime number.”

Factors are numbers we multiply together to produce another number. Here’s an example. Two times three is six. Two and three are factors of six. And prime number is a whole number greater than one that has exactly two factors: one and itself. For example, the number seven, the only two numbers that multiply together to equal seven are one and seven. So seven is a prime number. Then, we say that prime factorization is writing a composite number as the product of only prime numbers. Just a reminder, a composite number is a whole number greater than one that has more than two factors. For example, 10, you can find 10 by multiplying one by 10 or two by five. 10 has more than two factors.

Okay, back to prime factorization. We’re going to take these composite numbers and write them as a product of prime numbers. Remember that product means numbers that are multiplied together. We’re gonna look at two methods for finding prime factorization: the first one is by using a factor tree and the second one is dividing by prime numbers.

Find the prime factorization of 60.

Let’s start by using the factor tree method to factor 60. What you wanna do here is think of two numbers that you know that multiply together to equal 60. I’m gonna choose six and 10. Six times 10 equals 60. Now, we look at the numbers six and 10 and we see what multiplies together to equal each of those. For six, I know that two times three equals six. And for 10, two times five equals 10.

The factor tree method is finished when each of your branches is a prime number. So here, the only things that multiply together to equal two are two and one. Two is a prime number. Three is a prime number for the same reason: one times three is three. That’s the only factors. When we find that all of our branches have prime numbers, we can stop and write down the prime factorization. Two times two times three times five is the prime factorization of 60. 60 equals two times two times three times five. It’s often helpful to write it with exponents. So we say two squared times three times five for clarity. And we call this method the factor tree method. Let’s solve the same problem again using divide by prime numbers method.

For this method, I need to think of a prime number that 60 is divisible by. I’m going to start with two. I know that 60 is divisible by two. And I know that two is a prime number. 60 divided by two is 30. Now, I need a prime number here that 30 is divisible by. I chose three. 30 divided by three is 10. We need a prime number that 10 is divisible by. I’m gonna choose two. 10 divided by two is five. This method is finished when the number in the box is a prime number. Since five is a prime number, we’re finished with this step. We take all the prime numbers we’ve been using to divide and the bottom number. And this is the prime factorization.

You would say 60 equals two times three times two times five. Or more simply, two squared times three times five. Both methods lead us to the prime factorization of sixty. Though the methods are different, there is only one prime factorization of 60. Only one set of prime numbers multiply together to equal 60.

Our next example says, “Find all the prime factors of 28.”

This time we’re gonna try and divide by prime factors to solve the problem. I noticed that 28 is an even number. So I’m gonna start by dividing it by two. 28 divided by two leaves us with 14. Another even number! I’m gonna divide by two again. 14 divided by two is seven. We now recognize that seven is a prime number. And so that’s the end of this step. 28 equals two times two times seven. And we prefer to write it with exponents which leaves us with 28 equals two squared times seven.

Our next example: Find the prime factorization of 468.

You might be thinking that 468 is a really big number. Is it going to be super hard to find the prime factorization for 468? But the answer is no. We’re going to follow the same procedure. And it will be no different than finding the prime factorization for other numbers, other smaller numbers. Okay! Then let’s use the factor tree method and find the prime factorization here. Again, we recognize that this is an even number. So you can automatically start with two. Two times 234 equals 468. Two is a prime number. So this branch is finished.

234 is an even number. Let’s divide 234 by two. Two times 117 equals 234. Two is a prime number. This branch is finished. Now we need some factors of 117. If you don’t immediately recognize what the factors of 117 are, the best thing to do is try to check common prime factors. For example, we know that this is an odd number and not divisible by two. The sum of the digits one, one, seven is nine, which makes this divisible by three.

So our next step would be to divide 117 by three. Three times 39 equals 117. Three is a prime number. This branch is finished. I’m gonna move the thirty-nine over here to give us a little bit more space. What two factors multiply together to equal 39? It’s pretty easy to spot that 39 is divisible by three. 39 divided by 13 is three. Three is a prime number. This branch is finished. 13 is also prime, which means we’re finally down to all prime factors.

The end of each branch is a prime factor. We’re gonna put them all together to make the prime factorization. We get 468 is equal to two times two times three times three times 13, for a final answer with exponents of two squared times three squared times 13 equals 468. And it’s not so bad after all.

And finally, here are just a few tips for finding prime factorization. Number one: use the method that works for you. If you prefer dividing by prime factors, that’s great! You can find the answer by dividing by prime factors. If you prefer using the factor tree and that works for you, use that method. And secondly, if you get stuck and you’re not sure what factors to check for, check for larger prime numbers. Try dividing the number you’re factoring by 13, 17, 19, or other prime numbers. And third, just practice. Recognizing what factors are in numbers comes with practice. You’ll get faster and more accurate at checking for factorization with practice.

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