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