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. A 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 ten, you can find ten by multiplying one by ten or two by five. ten 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 sixty. Let’s start by using the factor tree method to factor sixty. What you wanna do here is think of two numbers that you know that multiply together to equal sixty. I’m gonna choose six and ten; six times ten equals sixty. Now we look at the numbers six and ten and we see what multiplies together to equal each of those. For six, I know that two times three equals six. And for ten, two times five equals ten. 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 sixty; sixty 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 sixty is divisible by. I’m going to start with two. I know that sixty is divisible by two, and I know that two is a prime number. sixty divided by two is thirty. Now I need a prime number here that thirty is divisible by. I chose three; thirty divided by three is ten. We need a prime number that ten is divisible by. I’m gonna choose two; ten 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 sixty 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 sixty. Only one set of prime numbers multiply together to equal sixty.
Our next example says find all the prime factors of twenty-eight. This time we’re gonna try and divide by prime factors to solve the problem. I noticed that twenty-eight is an even number so I’m gonna start by dividing it by two twenty-eight divided by two leaves us with fourteen. Another even number! I’m gonna divide by two again. fourteen divided by two is seven. We now recognize that seven is a prime number, and so that’s the end of this step. twenty-eight equals two times two times seven. And we prefer to write it with exponents which leaves us with twenty-eight equals two squared times seven.
Our next example, find the prime factorization of four hundred and sixty-eight. You might be thinking that four hundred and sixty-eight is a really big number. Is it going to be super hard to find the prime factorization for four hundred and sixty-eight? 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 recognise that this is an even number so you can automatically start with two. two times two hundred and thirty-four equals four hundred and sixty-eight. two is a prime number, so this branch is finished. two hundred and thirty-four is an even number. Let’s divide two hundred and thirty-four by two. two times one hundred and seventeen equals two hundred and thirty-four. two is a prime number; this branch is finished. Now we need some factors of one hundred and seventeen. If you don’t immediately recognize what the factors of one hundred and seventeen 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 one hundred and seventeen by three. three times thirty-nine equals one hundred and seventeen. 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 thirty-nine? It’s pretty easy to spot that thirty-nine is divisible by three. thirty-nine divided by thirteen is three. three is a prime number; this branch is finished. thirteen 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 four hundred and sixty-eight is equal to two times two times three times three times thirteen, for a final answer with exponents of two squared times three squared times thirteen equals four hundred and sixty-eight. 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 thirteen, seventeen, nineteen, 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.