Math Antics - Prime Factorization

Summary

This Math Antics video introduces prime factorization, explaining that prime numbers have only two factors: 1 and themselves. It lists prime numbers less than 20 and clarifies why 1 is not prime. The video demonstrates how composite numbers are made by multiplying primes together, using examples like 4 (2x2) and 6 (2x3). It then shows how to find the prime factorization of a number, like 12, using a factor tree to ensure all factors are prime, resulting in 2x2x3. The process is repeated for 42, leading to the prime factors 2, 3, and 7.

Takeaways

• 🔢 Prime factorization is the process of breaking down a composite number into its prime factors.
• 📐 A prime number is a number greater than 1 that has no divisors other than 1 and itself.
• 📋 The first few prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19.
• 🚫 The number 1 is not considered a prime number due to technical reasons, despite being similar to prime numbers.
• 🏗 Composite numbers are made up of prime numbers, which are their building blocks.
• 🌳 Prime factorization can be visualized using a 'factor tree', which helps track multiple factoring steps.
• 🔄 When factoring, continue until all factors are prime; this ensures the complete prime factorization of a number.
• 🔄 The order in which you start factoring does not affect the final set of prime factors obtained.
• 🔄 For example, the prime factorization of 12 can be found as 2 × 2 × 3, regardless of starting with 2 × 6 or 4 × 3.
• 🔄 Similarly, the prime factorization of 42 is 2 × 3 × 7, starting with dividing by 2 and then factoring the result.
• 📝 Practice is essential to become proficient in finding prime factorizations of composite numbers.

Q & A

What is prime factorization?

Prime factorization is the process of breaking down a composite number into its prime factors, which are the prime numbers that can be multiplied together to produce the original number.

What is a prime number?

A prime number is a whole number greater than 1 that has exactly two distinct positive divisors: 1 and itself.

Why isn't 1 considered a prime number?

1 is not considered a prime number because it only has one distinct positive divisor, which is itself, and the definition of a prime number requires two distinct positive divisors.

What is the smallest prime number?

The smallest prime number is 2, which is the only even prime number.

How can you determine if a number is prime?

To determine if a number is prime, you can test for divisibility by all prime numbers less than its square root. If none of these primes divide the number without a remainder, then the number is prime.

What is a composite number?

A composite number is a positive integer greater than 1 that has more than two distinct positive divisors, meaning it can be factored into a product of smaller natural numbers.

How does prime factorization relate to composite numbers?

Prime factorization is the process used to express a composite number as a product of prime numbers, showing the fundamental building blocks of the composite number.

What is a factor tree?

A factor tree is a diagram used to represent the prime factorization of a number. It shows the steps of factoring a number into prime factors by branching out from the original number to its prime factors.

Why does the order of factoring matter in prime factorization?

The order of factoring does not change the final set of prime factors. Regardless of the order in which the factoring is done, the same prime factors will be obtained as long as the process is completed down to prime numbers.

How can you practice prime factorization?

You can practice prime factorization by attempting to factorize various composite numbers into their prime factors, using either a factor tree or by testing for divisibility by prime numbers.

What is the prime factorization of 12?

The prime factorization of 12 is 2 × 2 × 3, which can also be written as 2^2 × 3.

How does the process of prime factorization help in understanding numbers?

Prime factorization helps in understanding numbers by revealing their fundamental building blocks, which can be useful in various mathematical applications such as solving Diophantine equations and in number theory.

Outlines

00:00

🔢 Introduction to Prime Factorization

The video introduces the concept of prime factorization, starting with the definition of prime numbers. Prime numbers are those that have exactly two distinct positive divisors: 1 and the number itself. The script uses the number 7 as an example to demonstrate that it can only be divided by 1 and itself without a remainder, thus identifying it as a prime number. A list of prime numbers less than 20 is provided, and the distinction between prime and composite numbers is explained. Composite numbers are those that can be factored into prime numbers. The script then explains how composite numbers are made by multiplying different combinations of prime numbers, and introduces the term 'prime factorization' both as a set of prime factors and as the action of finding these factors. The process of prime factorization is illustrated using a 'factor tree' with the example of the number 12, which is factored into prime numbers 2, 2, and 3.

05:02

🌿 Consistency in Prime Factorization

This paragraph further explores prime factorization by emphasizing that no matter how you start the factoring process, as long as you continue until all factors are prime, you will end up with the same set of prime factors. The script uses the number 42 as an example to demonstrate this concept. It shows that 42 can be factored into 2 × 21, and then further into 2 × 3 × 7, since 21 is factored into 3 × 7, and both 2, 3, and 7 are prime numbers. The video concludes by encouraging viewers to practice prime factorization to become proficient in the skill, and directs them to the website www.mathantics.com for more information and practice exercises.

Keywords

💡Prime Factorization

Prime factorization is the process of breaking down a composite number into its smallest prime factors. It's a fundamental concept in number theory and is central to the video's theme. The script uses prime factorization to explain how composite numbers are built from prime numbers. For example, the number 12 is factored into 2 × 2 × 3, showing that it's composed of prime numbers.

💡Prime Number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Prime numbers are the building blocks of all other numbers, as they cannot be formed by multiplying other numbers together. The script lists prime numbers less than 20 and uses them to demonstrate the concept of prime factorization.

💡Composite Number

A composite number is a positive integer that has at least one positive divisor other than one or itself, meaning it can be formed by multiplying two smaller natural numbers. The video explains that composite numbers are made up of prime numbers, as shown by the example of the number 4, which is the product of 2 × 2.

💡Factoring

Factoring refers to the process of breaking down a number into its factors, which are the numbers that can be multiplied together to produce the original number. The script discusses factoring in the context of prime factorization, emphasizing that the process continues until all factors are prime.

💡Divisibility

Divisibility is a property of integers that describes the ability of one number to be divided by another without leaving a remainder. The script uses the concept of divisibility to test whether numbers are prime or composite by checking for factors other than 1 and the number itself.

💡Factor Tree

A factor tree is a diagram used to represent the factorization of a number, showing the steps of breaking down the number into its prime factors. The video uses a factor tree to illustrate the process of prime factorization for the number 12, showing how it can be broken down into 2 × 2 × 3.

💡Multiplication

Multiplication is the mathematical operation of scaling one number by another, resulting in the product of the two. In the context of the video, multiplication is used to demonstrate how prime numbers can be combined to form composite numbers, as in the example of 2 × 3 to get 6.

💡Remainder

A remainder is what is left over after division of one number by another. The script uses the absence of a remainder as a test for divisibility, which is a clue to whether a number might be prime, as in the case of the number 7, which only has 1 and itself as factors.

💡Building Blocks

In the video, 'building blocks' is a metaphor for prime numbers, suggesting that they are the fundamental components from which all other numbers are constructed. This concept is used to explain the importance of prime numbers in mathematics.

💡Composite

The term 'composite' is used to describe a number that is not prime, meaning it can be broken down into smaller factors. The script explains that composite numbers are composed of prime numbers, which is why finding their prime factorization is important.

💡Exercise

The video script encourages viewers to practice what they've learned through exercises, which is a common method for reinforcing mathematical concepts. The exercises related to prime factorization help viewers apply the concepts discussed in the video.

Highlights

Introduction to prime factorization.

Prime factorization involves factoring numbers into primes.

Prime numbers are numbers that have exactly two factors: 1 and itself.

List of prime numbers less than 20.

Explanation of why 1 is not considered a prime number.

Prime numbers are the building blocks of all whole numbers.

Composite numbers are made by multiplying prime numbers together.

Prime factorization is both a set of prime factors and an action to find them.

Demonstration of prime factorization using a factor tree.

Prime factorization of 12 using a factor tree.

Explaining that all factorizations of 12 lead to the same prime factors.

Prime factorization of 42 using divisibility tests.

Prime factorization of 42 results in 2 × 3 × 7.

Encouragement to practice prime factorization.

The video concludes with a reminder to visit www.mathantics.com for more information.

Transcripts

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.

Learn more at www.mathantics.com