Math Antics - Prime Factorization

mathantics2012-04-16

Math#educational#Math Antics#arithmetic#multi-digit#addition#subtraction#multiplication#division#prime#prime numbers#composite#composite numbers#prime factorization#factor tree#whole numbers

2M views|12 years ago

💫 Short Summary

The video explains prime factorization using the example of finding the prime factorization of 12, demonstrates the use of a factor tree to break down composite numbers into their prime factors, and concludes by showing that no matter which way you factor a number, you will always end up with the same group of prime factors.

✨ Highlights

📊 Transcript

✦

Introduction to prime factorization and prime numbers.

00:06

Prime factorization involves factoring a number using prime numbers.

Prime numbers have exactly two factors: 1 and the number itself.

✦

Composite numbers are the product of prime numbers.

02:25

Composite numbers are formed by multiplying prime numbers together.

Each combination of prime numbers for a composite number is called its prime factorization.

✦

Using a factor tree to find the prime factorization of a number.

03:20

Prime factorization is the set of prime numbers multiplied together to get another number.

Factor tree is a diagram that helps track multiple factoring steps.

✦

Prime factorization of 12 is 2 x 2 x 3.

04:30

Factor tree for 12: 2 x 6, then 2 x 3.

Another way to factor 12: 4 x 3, then 2 x 2 x 3.

No matter which way you factor, you'll always end up with the same prime factors.

✦

Finding the prime factorization of 42.

05:14

42 is factored into 2 x 21, then 3 x 7.

Prime factorization of 42 is 2 x 3 x 7.

00:06Hi and welcome to Math Antics.

00:08In this video, we’re gonna learn something called ‘prime factorization’.

00:12Wow! That sounds pretty complicated, doesn’t it? But don’t worry, it’s not that bad.

00:18Now from the name “prime factorization”, you can probably guess that it involves factoring like we learned about in the last video.

00:25But what about this word “prime” here? What does that mean?

00:28Well, to help you understand that, let’s use what we learned in the last video to factor the number 7.

00:34Well 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,

00:41it’s about what you can MULTIPLY.

00:43Well, since I can’t think of any numbers that would work, let’s find factors by ‘testing for divisibility’.

00:48Now I’m going to do this really fast using my calculator.

00:52Let’s see…

00:53[morse code beeps]

00:58Okay, here’s the numbers I got.

01:00That’s interesting… the only two numbers that didn’t leave remainders were 1 and 7. And those are kinda obvious!

01:07We know that if you multiply ANY number by 1, you’ll just get that same number.

01:11But why aren’t there any OTHER factors of 7?

01:15Alright, here’s why… 7 is a special kind of number called a PRIME number.

01:20Now a prime numbers is just a number that has exactly two factors: itself and 1.

01:26There’s a lot of prime numbers.

01:28Here’s a list of all the prime numbers that are less than 20:

01:312, 3, 5, 7, 11, 13, 17 and 19.

01:36They’re the one’s you’ll use most often.

01:38Now some of you might be wondering why 1 isn’t on the list of prime numbers.

01:42Well, 1 is a lot like a prime number, but for some technical reasons, it’s not considered prime.

01:48Okay, so in a way, prime numbers are just special numbers that you can’t factor.

01:53Well, unless you use the obvious factors of 1 and the number itself.

01:57But what’s so special about prime numbers anyway? Why do we need to know about them?

02:02Well, prime numbers are like the building blocks of all the other whole numbers.

02:06In fact, whole numbers that are not prime are called ‘composite’ numbers because they’re composed of primes.

02:12That means that you can get them by multiplying prime numbers together.

02:16Here’s a good way to see how that works.

02:18Again we’ll list all the prime numbers that are less than 20.

02:21And now let’s list at all the composite numbers that are less than 20 over here.

02:25The 1st composite number is 4, and you get 4 by multiplying the primes 2 × 2

02:31The next composite number is 6, and you get it by multiplying the primes 2 × 3

02:36And the next composite number is 8, which you get by multiplying the primes 2 × 2 × 2

02:42And the composite number 9 can be made by multiplying the primes 3 × 3

02:47We could keep going like this and you would see that ALL the composite numbers are made by multiplying different combinations of prime numbers together.

02:55And each of these combinations is called the “prime factorization” of its composite number.

03:01AH… so the ‘prime factorization’ is a THING.

03:04It’s the set of prime numbers that you multiply together to get another number.

03:08That’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.

03:17And that’s what we’re gonna do next.

03:20We’re gonna use prime factorization (the action) to find the prime factorization (the set of prime factors) for the number 12.

03:28And that just means that we’ll continue to factor 12 until all the factors are prime numbers.

03:34Now to do this, I’m gonna use something called a ‘factor tree’.

03:37A factor tree is just a diagram that helps you keep track of multiple factoring steps.

03:42When you factor a number, you write the two factors below it with lines (or branches) going to them.

03:48And then, if you factor one of the factors, you do the same thing again.

03:52You’ll see how it works as we do this example, so let’s get started.

03:5612 can be factored into 2 × 6. So we’re done, right? Well not yet.

04:02Because we’re doing PRIME factorization, we need to keep going until all the factors are prime numbers.

04:07So let’s see if they are.

04:09Well we know that 2 is a prime number, but is 6 prime?

04:12No it’s not because 6 can be factored into 2 × 3.

04:16And both 2 and 3 ARE prime numbers, so now we’re done factoring.

04:21And if we bring down that 2 that we had from the first factoring step,

04:24we can see that the prime factorization of 12 is 2 × 2 × 3.

04:30Now I know what some of you are thinking.

04:32“I didn’t want to factor 12 into 2 × 6. I wanted to factor it into 4 × 3.”

04:37Well okay then, let’s try it that way.

04:40This time we’ll start by factoring 12 into 4 × 3.

04:44But remember, we need to keep factoring until all our factors are prime numbers.

04:48So let’s see… are 4 and 3 prime numbers?

04:51Well, 3 is prime, but 4 is not. 4 can be factored into the primes 2 × 2.

04:56And again, if we bring down that 3 from the first step, we see that we have 3 prime factors for 12,

05:02and they’re the EXACT same ones that we got the first time.

05:05That means, no matter which way you start factoring,

05:08as long as you factor all the way down to prime numbers, you’ll always end up with the same group of prime factors.

05:14Let’s try just one more to make sure you’ve got it. Let’s find the prime factorization of 42.

05:20Well, for the first step of our factoring, I see that 42 is an even number,

05:24so that means that we can divide it by 2 to get our first 2 factors.

05:28So 42 divided by 2 equals 21, so we can factor 42 into 2 × 21.

05:35Okay, 2 is prime, so we can’t factor it anymore. But what about 21?

05:39Well, if you’ve memorized your multiplication table, you might recognize that 21 is one of the answers on it.

05:45You can get 21 by multiplying 3 × 7, so we can factor 21 into 3 × 7.

05:51And if you didn’t remember that, you could have just done some divisibility tests and you would have figured it out.

05:56Okay, so what about the 3 and 7? Well, they’re both prime, so that means that we're done factoring.

06:03We bring down the 2 from the first step and we can see that the prime factorization of 42 is 2 × 3 × 7.

06:11Alright… now you know what prime numbers are,

06:13and you know how to use prime factorization to find the set of prime factors that a composite number is made of.

06:19And that set of numbers is ALSO called its prime factorization. (just to confuse you)

06:25As usual, it’s important to practice what you’ve learned in this video so that you’ll get good at it.

06:29And you can practice by doing the exercises for this section.

06:32Good luck and thanks for watching Math Antics. See you next time.

06:37Learn more at www.mathantics.com

💫 FAQs about This YouTube Video

1. What is prime factorization and how is it related to prime numbers?

Prime factorization is the process of breaking down a number into a set of prime numbers that, when multiplied together, give the original number. This is related to prime numbers because prime factorization involves finding the prime numbers that are the building blocks of the original number. Prime numbers, by definition, have only two factors: 1 and the number itself, and they are the fundamental components used in prime factorization to express composite numbers.

2. How are prime numbers important in mathematics?

Prime numbers are important in mathematics because they are the building blocks of all other whole numbers. Any non-prime (composite) number can be expressed as a product of prime numbers. This property of prime numbers is crucial in various mathematical calculations, including simplifying fractions, finding the greatest common divisor, and RSA encryption in the field of cryptography.

3. What is a composite number and how is it related to prime factorization?

A composite number is a positive integer greater than 1 that has more than two positive divisors. In other words, a composite number is a number that is not prime. Prime factorization is the process of expressing a composite number as the product of prime numbers, emphasizing the relationship between composite numbers and the prime numbers that form their prime factorization.

4. Can you explain the concept of prime factorization with an example?

Certainly! Prime factorization is best explained with an example. Let's take the number 24. The prime factorization of 24 is 2 x 2 x 2 x 3, which means 2 and 3 are the prime numbers that, when multiplied together, give the original number 24. This demonstrates how prime factorization breaks down a composite number into its prime components.

5. How do factor trees help in finding the prime factorization of a number?

Factor trees are a visual tool that aid in finding the prime factorization of a number by systematically breaking it down into its prime factors. The number is split into two factors, and each factor is further split until only prime factors are left. The prime factorization is then obtained by multiplying all the prime factors together. Factor trees provide a methodical and organized approach to prime factorization.

6. Why is it important to understand prime factorization in mathematics?

Understanding prime factorization is important in mathematics because it provides insights into the structure of numbers. It simplifies various mathematical operations such as finding greatest common divisors, determining divisibility, and simplifying fractions. Prime factorization also plays a crucial role in advanced mathematical concepts and real-world applications, highlighting its significance in mathematical understanding and problem-solving.

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