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Prime factorization, this is a hugely important topic. First of all, here are the first 17 prime numbers, all the primes from two up to just below 60. And of course, the first eight, those are nonnegotiable, you absolutely need to. To know those, and the others are very good ones to know, as well. In the last video, I suggested that prime numbers were the building blocks of all the positive integers.
Now we can discuss what this means. In order to talk about this, I’m going to introduce the name of an important mathematical theorem, The Fundamental Theorem of Arithmetic. I’ll say right now, you don’t need to know this particular theorem. As it was happened, it was first discussed by Mr. Euclid. You don’t need to know that, but you do need to know the idea of this theorem.
Every positive integer greater than 1 must either be, A, a prime number, or B, be expressed as a unique product of prime numbers. So in other words, every integer greater than 1 that’s not prime can be expressed as a, as a product of primes and this product is called the prime factorization of the number.
So that core idea you definitely need to know. Let’s talk about this prime factorization. Let’s give some examples. So first of all, 9 is obviously 3 times 3. It’s a product of two prime numbers.
10 is 2 times 5. That’s a product of two prime numbers. 12, we could write that as 4 times 3 but 4 isn’t a prime number. 4 itself, we can write it as 2 times 2. So we can write 12 as 2 times 2 times 3. That’s a product of prime numbers.
15 is 3 times 5, 24 we could write that as 8 times 3. Well, 8 is not a prime number, we can write the 8 as 2 times 4, 4 is not a prime number, we can write that as 2 times 2. If we had realized 8 is 2 cubed, we could’ve skipped directly ahead to say that 8 is 2 times 2 times 2. Well, 2 times 2 times 2 times 3 is the prime factorization in 24.
In fact, we can even write this, this way with an exponent, 2 to the 3rd times 3. Either way is correct to write it. When the numbers tend to get larger and you have lots and lots of prime factors, it’s usually more convenient to write it in exponent form. Simply because it’s more compact. So, here’s some larger numbers.
Now 100, you might be able to see 100 is 10 times 10. Each 10 is 2 times 5. So, that’s how we get 2 cubed, 2 squared times 5 squared. Here’s some other ones that are a bit larger. You don’t necessarily need to worry right now about how we found these. These are just to show you some examples of prime factorizations.
Let’s talk about how to find the prime factorization of a slightly larger number. So, for example, suppose we’re looking at the number 96. What is the prime factorization of 96? You might wanna pause the video and work on this right now, and then I’ll discuss it. So the prime factorization of 96.
Well, certainly is divisible by two. So we can express it as 2 times 48. 48 is 6 times 8. Well, from there, 6 is 2 times 3, that’s the prime factorization. 8 is 2 times 2 times 2, then group everything together. We have one, two, three, four, five factors of two so we can write that as 2 to the 5th times 3.
That’s the prime factorization of 96. Notice that 96 would be divisible by two, four, six, two, four, eight, and 16. Those are powers of 2 and by 6, 12, 24 and 48, those are those four powers of 2 multiplied by 3, all of those would have to be factors of 96. Notice that it would not be divisible by 15 or 18. Not divisible by 15 because 15 involves the factor of 5.
And there’s obviously no factor of five in 96. 18 is a little trickier. 18 is 2 times 3 times 3. You need two factors of three. 96 only has one factor of three, and therefore 96 cannot possibly be divisible by 18.
So, here I’ve just shown an example of a very important idea. One way to say it is the following. The prime factorization of a number is like the DNA of the number, revealing all its essential ingredients. Any factor of Q must be composed only of prime factors found in Q. So that’s a tricky idea.
We’re gonna talk about this for a few minutes because its a tricky idea. First thing I’ll say is remember our synonymous statements from some earlier videos. We have these four statements, r is a factor of Q, r is a divisor of Q, Q is divisible by r, Q is a multiple of r. All four of those mean the same thing.
Well, we can add a fifth statement now. Every prime factor of r is included in the prime factorization of Q. That is equivalent to the other four statements. Let’s talk about an example with actual numbers. Suppose we are told 4680 has this prime factorization. Someone just hands us the prime factorization, which is awfully nice.
Given this prime factorization we want to know which of the following numbers are or aren’t factors in this number. So, I’m going to say pause the video and see if you can use the prime factorization to figure out whether each number here is a factor of that number. So for 25, 25 is obviously 5 time 5, we need two factors of 5.
We have only one factor of 5, so that’s not gonna work. 25 does not work. 45, well that’s 5 times 9, or 5 times 3 times 3. Well, we have a factor of 5 and we have two factors of 3, so that works. 65 is 5 times 13. Well, we have a 5.
We have a 13. So, that works. 85 is 5 times 17. We do not have a factor of 17. So, that’s not gonna work. 120, well, that is 40 times 3.
40 is 8 times 5. So we have a 3 and a 5 and then a 8 which is 2 times 2 times 2. Well, we have a 2 cubed, we have a 3 and we have a 5. So that, we’re all set means that this would be divisible by 120. 180, well that is 30 times 6, 6 is 2 times 3. 30 is 6 times 5.
So we get 2 times 3 times 5, so we need two powers of 3, two powers of 2, and one power of 5. And we have all that in the prime factorization of 4,680. So in other words, 4,680 would be divisble by 180. That would actually work. So we’ve used the prime factorization to figure out whether these other numbers could be divisors of 4680.
That’s just one example of how prime factorizations can be useful. In the upcoming videos, we’ll be showing many, many different examples of how we can use prime factorizations to answer a number of different integer property questions. Finding the prime factorization of a number is definitely the single most important skill in the entire integer properties section.
It’s critically important to practice this skill until you are extremely comfortable with it. In summary, we talked about what the prime factorization is and how to find it. And we talked about how the idea of prime factorization connects with the ideas of factors and multiples.
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