Full transcript
0:06Hi and welcome to Math Antics.
0:08In this video, we’re gonna learn something called ‘prime factorization’.
0:12Wow! That sounds pretty complicated, doesn’t it? But don’t worry, it’s not that bad.
0:18Now from the name “prime factorization”, you can probably guess that it involves factoring like we learned about in the last video.
0:25But what about this word “prime” here? What does that mean?
0:28Well, to help you understand that, let’s use what we learned in the last video to factor the number 7.
0:34Well let’s see… We could get 7 by adding 3 and 4, but factoring’s not about what you can ADD to get a number,
0:41it’s about what you can MULTIPLY.
0:43Well, since I can’t think of any numbers that would work, let’s find factors by ‘testing for divisibility’.
0:48Now I’m going to do this really fast using my calculator.
0:52Let’s see…
0:53[morse code beeps]
0:58Okay, here’s the numbers I got.
1:00That’s interesting… the only two numbers that didn’t leave remainders were 1 and 7. And those are kinda obvious!
1:07We know that if you multiply ANY number by 1, you’ll just get that same number.
1:11But why aren’t there any OTHER factors of 7?
1:15Alright, here’s why… 7 is a special kind of number called a PRIME number.
1:20Now a prime numbers is just a number that has exactly two factors: itself and 1.
1:26There’s a lot of prime numbers.
1:28Here’s a list of all the prime numbers that are less than 20:
1:312, 3, 5, 7, 11, 13, 17 and 19.
1:36They’re the one’s you’ll use most often.
1:38Now some of you might be wondering why 1 isn’t on the list of prime numbers.
1:42Well, 1 is a lot like a prime number, but for some technical reasons, it’s not considered prime.
1:48Okay, so in a way, prime numbers are just special numbers that you can’t factor.
1:53Well, unless you use the obvious factors of 1 and the number itself.
1:57But what’s so special about prime numbers anyway? Why do we need to know about them?
2:02Well, prime numbers are like the building blocks of all the other whole numbers.
2:06In fact, whole numbers that are not prime are called ‘composite’ numbers because they’re composed of primes.
2:12That means that you can get them by multiplying prime numbers together.
2:16Here’s a good way to see how that works.
2:18Again we’ll list all the prime numbers that are less than 20.
2:21And now let’s list at all the composite numbers that are less than 20 over here.
2:25The 1st composite number is 4, and you get 4 by multiplying the primes 2 × 2
2:31The next composite number is 6, and you get it by multiplying the primes 2 × 3
2:36And the next composite number is 8, which you get by multiplying the primes 2 × 2 × 2
2:42And the composite number 9 can be made by multiplying the primes 3 × 3
2:47We could keep going like this and you would see that ALL the composite numbers are made by multiplying different combinations of prime numbers together.
2:55And each of these combinations is called the “prime factorization” of its composite number.
3:01AH… so the ‘prime factorization’ is a THING.
3:04It’s the set of prime numbers that you multiply together to get another number.
3:08That’s true, but you can also use the term ‘prime factorization’ as an ACTION to describe how we find out what prime numbers a composite number is made of.
3:17And that’s what we’re gonna do next.
3:20We’re gonna use prime factorization (the action) to find the prime factorization (the set of prime factors) for the number 12.
3:28And that just means that we’ll continue to factor 12 until all the factors are prime numbers.
3:34Now to do this, I’m gonna use something called a ‘factor tree’.
3:37A factor tree is just a diagram that helps you keep track of multiple factoring steps.
3:42When you factor a number, you write the two factors below it with lines (or branches) going to them.
3:48And then, if you factor one of the factors, you do the same thing again.
3:52You’ll see how it works as we do this example, so let’s get started.
3:5612 can be factored into 2 × 6. So we’re done, right? Well not yet.
4:02Because we’re doing PRIME factorization, we need to keep going until all the factors are prime numbers.
4:07So let’s see if they are.
4:09Well we know that 2 is a prime number, but is 6 prime?
4:12No it’s not because 6 can be factored into 2 × 3.
4:16And both 2 and 3 ARE prime numbers, so now we’re done factoring.
4:21And if we bring down that 2 that we had from the first factoring step,
4:24we can see that the prime factorization of 12 is 2 × 2 × 3.
4:30Now I know what some of you are thinking.
4:32“I didn’t want to factor 12 into 2 × 6. I wanted to factor it into 4 × 3.”
4:37Well okay then, let’s try it that way.
4:40This time we’ll start by factoring 12 into 4 × 3.
4:44But remember, we need to keep factoring until all our factors are prime numbers.
4:48So let’s see… are 4 and 3 prime numbers?
4:51Well, 3 is prime, but 4 is not. 4 can be factored into the primes 2 × 2.
4:56And again, if we bring down that 3 from the first step, we see that we have 3 prime factors for 12,
5:02and they’re the EXACT same ones that we got the first time.
5:05That means, no matter which way you start factoring,
5:08as long as you factor all the way down to prime numbers, you’ll always end up with the same group of prime factors.
5:14Let’s try just one more to make sure you’ve got it. Let’s find the prime factorization of 42.
5:20Well, for the first step of our factoring, I see that 42 is an even number,
5:24so that means that we can divide it by 2 to get our first 2 factors.
5:28So 42 divided by 2 equals 21, so we can factor 42 into 2 × 21.
5:35Okay, 2 is prime, so we can’t factor it anymore. But what about 21?
5:39Well, if you’ve memorized your multiplication table, you might recognize that 21 is one of the answers on it.
5:45You can get 21 by multiplying 3 × 7, so we can factor 21 into 3 × 7.
5:51And if you didn’t remember that, you could have just done some divisibility tests and you would have figured it out.
5:56Okay, so what about the 3 and 7? Well, they’re both prime, so that means that we're done factoring.
6:03We bring down the 2 from the first step and we can see that the prime factorization of 42 is 2 × 3 × 7.
6:11Alright… now you know what prime numbers are,
6:13and you know how to use prime factorization to find the set of prime factors that a composite number is made of.
6:19And that set of numbers is ALSO called its prime factorization. (just to confuse you)
6:25As usual, it’s important to practice what you’ve learned in this video so that you’ll get good at it.
6:29And you can practice by doing the exercises for this section.
6:32Good luck and thanks for watching Math Antics. See you next time.
6:37Learn more at www.mathantics.com