Prime numbers | Factors and multiples | Pre-Algebra | Khan Academy

Summary

This video introduces the concept of prime numbers, explaining that a prime number is a natural number divisible by exactly two natural numbers: itself and 1. Through several examples, the video illustrates which numbers are prime and which are not, highlighting how prime numbers serve as the 'building blocks' of numbers, similar to atoms in physics. The video also touches on how primes are important in areas like cryptography. Examples include the numbers 2, 3, 5, and 7, while larger numbers like 16 and 51 are shown to not be prime.

Takeaways

• 🔢 Prime numbers are natural numbers divisible by exactly two numbers: 1 and itself.
• 🧐 The smallest natural number, 1, is not a prime number because it is divisible by only one number, itself.
• ✔️ The number 2 is the only even prime number since it's divisible only by 1 and 2.
• 🔎 The number 3 is also a prime number because it is divisible only by 1 and 3.
• ❌ The number 4 is not prime because it is divisible by 1, 2, and 4.
• ✅ The number 5 is prime since it can only be divided by 1 and 5.
• 🔄 Numbers divisible by more than two natural numbers, like 6, are not prime.
• 💡 Prime numbers are building blocks of numbers, similar to how atoms are building blocks of matter.
• ❓ Larger numbers can be prime, like 17, which is divisible only by 1 and 17.
• ❌ The number 51 is not prime as it is divisible by 3 and 17, not just 1 and 51.

Q & A

What is the basic definition of a prime number?

A prime number is a natural number that is divisible by exactly two natural numbers: itself and 1.

Why is 1 not considered a prime number?

Although 1 is divisible by itself, it is only divisible by one natural number, not two, which is required for a number to be prime.

Why is 2 considered a special prime number?

2 is the only even prime number because it is divisible only by 1 and 2, while all other even numbers are divisible by 2 and other numbers.

Why is 4 not a prime number?

4 is divisible by 1, 2, and 4, meaning it has more than two divisors, which disqualifies it as a prime number.

How can prime numbers be thought of as the building blocks of numbers?

Prime numbers are similar to atoms in that they cannot be broken down into products of smaller natural numbers. Non-prime numbers, like 6, can be factored into primes (e.g., 6 = 2 x 3).

What makes 7 a prime number?

7 is divisible only by 1 and 7, and no other numbers, making it a prime number.

What is an example of a non-prime number with multiple divisors?

16 is not a prime number because it is divisible by 1, 2, 4, 8, and 16, meaning it has multiple divisors beyond just 1 and itself.

Is 17 a prime number? Why or why not?

Yes, 17 is a prime number because it is divisible only by 1 and 17, with no other divisors.

What makes 51 a non-prime number?

51 is not prime because it is divisible by 3 and 17, in addition to 1 and 51.

What is the importance of prime numbers in fields like cryptography?

Prime numbers are fundamental in cryptography because they serve as the basis for encryption algorithms that protect information in computing systems.

Outlines

00:00

🔢 Introduction to Prime Numbers and Their Importance

In this paragraph, the video introduces the concept of prime numbers and their significance. The speaker mentions that prime numbers are foundational in mathematics and have practical applications in fields like cryptography, where encryption may rely on them. The definition of prime numbers is provided: a natural number divisible by exactly two numbers, itself and 1. This explanation is followed by examples, clarifying that numbers like 2 and 3 are prime, while 1 is not, as it doesn't meet the criteria of divisibility by two natural numbers.

05:01

📏 Examples of Prime Numbers and Their Divisibility

The video dives deeper into testing if numbers like 4, 5, 6, and 7 are prime. The method involves checking divisibility by numbers other than 1 and itself. For instance, 4 is not prime because it's divisible by 2, while 5 is prime since it's only divisible by 1 and 5. Similarly, 6 has more than two divisors, and 7 is prime. This pattern of checking divisibility continues, helping the viewer understand how prime numbers function and why certain numbers fail the prime test.

🧱 Prime Numbers as the Building Blocks of Math

Prime numbers are compared to the building blocks of mathematics, much like atoms in science. The speaker explains that primes can't be broken down further into smaller natural numbers, making them fundamental. Numbers like 6, which can be factored into primes (2 and 3), are examples of how non-prime numbers can be decomposed. This is contrasted with prime numbers like 7, which cannot be simplified beyond 1 and itself. The analogy emphasizes the critical role primes play in the structure of number theory.

🔍 Testing Larger Numbers for Primality

This paragraph explores the primality of larger numbers like 16, 17, and 51. The speaker demonstrates that while 16 has multiple factors (e.g., 2×8, 4×4), making it non-prime, 17 is prime because it is only divisible by 1 and 17. The number 51 is used as a trick question, with the speaker explaining that it is not prime because it can be divided by 3 and 17. This provides a more complex example to help viewers understand how to test numbers for primality.

Keywords

💡Prime Number

A prime number is defined as a natural number divisible by exactly two distinct natural numbers: 1 and itself. This concept is central to the video's theme, where the speaker explains how prime numbers are the building blocks of natural numbers, as they cannot be broken down further. For example, 2, 3, 5, and 7 are prime numbers because they meet this criterion.

💡Natural Number

Natural numbers are the positive integers starting from 1, used in counting (1, 2, 3, etc.). In the video, natural numbers are important because the definition of prime numbers specifically refers to divisibility by natural numbers. The speaker contrasts natural numbers with other types like negative numbers or fractions, which are not relevant to prime numbers.

💡Divisibility

Divisibility refers to the ability of one number to be evenly divided by another, resulting in an integer. This concept is crucial to determining whether a number is prime, as a prime number is divisible only by 1 and itself. For example, 2 is divisible by 1 and 2, making it prime, but 4 is divisible by 1, 2, and 4, making it non-prime.

💡Factors

Factors are the numbers that multiply together to produce another number. In the context of prime numbers, a prime has exactly two factors: 1 and itself. Numbers that have more than two factors, like 4 (with factors 1, 2, and 4), are not prime. Understanding factors is essential in recognizing the difference between prime and composite numbers.

💡Composite Number

A composite number is a natural number greater than 1 that has more than two factors. In the video, composite numbers like 4, 6, and 16 are used as examples to contrast with prime numbers. These numbers can be broken down into smaller factors, unlike primes, which are indivisible beyond 1 and themselves.

💡Cryptography

Cryptography is mentioned as a sophisticated concept built on prime numbers. It refers to the practice of secure communication, where prime numbers play a key role in encryption algorithms. The speaker hints that the computers we use today rely on prime numbers to encrypt data securely, though no detailed explanation is given in this particular video.

💡Encryption

Encryption is a process of encoding information to prevent unauthorized access, often using prime numbers as part of the algorithm. While the video doesn't delve into technical details, it emphasizes that prime numbers are foundational to modern encryption, underlining their importance beyond basic arithmetic.

💡Atom

In the video, prime numbers are metaphorically compared to atoms, which were once thought to be indivisible. Prime numbers, like atoms, cannot be broken down into smaller parts (other than 1 and the number itself). This comparison helps illustrate the fundamental role of primes in number theory.

💡Counting Numbers

Counting numbers refer to the sequence of natural numbers starting from 1, used in basic arithmetic. The video emphasizes that prime numbers are found among these counting numbers. For instance, when the speaker checks whether a number like 3, 5, or 7 is prime, they are evaluating it within the context of counting numbers.

💡Divisible by Two

Divisibility by two is a special case highlighted in the video, particularly in the context of even numbers. Any even number greater than 2 is not prime because it is divisible by 2, which adds another factor besides 1 and the number itself. The speaker notes that 2 is the only even prime number.

Highlights

Introduction to the concept of prime numbers and their relevance in mathematics.

Prime numbers have applications in cryptography and encryption, making them significant beyond basic math.

Definition of prime numbers: a natural number divisible by exactly two natural numbers—1 and itself.

Clarification that 1 is not considered a prime number because it is divisible by only one number, not two.

The number 2 is the only even prime number because all other even numbers are divisible by more than just 1 and themselves.

Prime numbers like 3, 5, and 7 follow the rule of divisibility by exactly two numbers: 1 and themselves.

Non-prime numbers (composite numbers) like 4, 6, and 16 can be broken down into smaller factors.

Prime numbers are described as the 'building blocks' of numbers, much like atoms are for matter.

You can break composite numbers down into products of smaller natural numbers, but prime numbers can't be factored further.

The distinction between composite numbers (e.g., 6 = 2 x 3) and prime numbers, which cannot be decomposed further.

Number 7 is an example of a prime because it can only be divided by 1 and itself.

Example of non-prime numbers: 16, which is divisible by 1, 2, 4, 8, and 16, disqualifying it from being prime.

Number 17 is highlighted as a prime number because it's only divisible by 1 and 17, no other factors.

The challenge of determining if larger numbers like 51 are prime, and explanation that 51 is not prime since it's divisible by 3 and 17.

Encouragement to practice identifying prime numbers through exercises and future videos.

Transcripts

In this video, I want to talk a little bit

about what it means to be a prime number.

And what you'll see in this video,

or you'll hopefully see in this video,

is it's a pretty straightforward concept.

But as you progress through your mathematical careers,

you'll see that there's actually fairly sophisticated concepts

that can be built on top of the idea of a prime number.

And that includes the idea of cryptography.

And maybe some of the encryption that your computer uses

right now could be based on prime numbers.

If you don't know what encryption means,

you don't have to worry about it right now.

You just need to know the prime numbers are pretty important.

So I'll give you a definition.

And the definition might be a little confusing,

but when we see it with examples,

it should hopefully be pretty straightforward.

So a number is prime if it is a natural number--

and a natural number, once again, just as an example,

these are like the numbers 1, 2, 3, so essentially the counting

numbers starting at 1, or you could

say the positive integers.

It is a natural number divisible by exactly two numbers,

or two other natural numbers.

Actually I shouldn't say two other,

I should say two natural numbers.

So it's not two other natural numbers--

divisible by exactly two natural numbers.

One of those numbers is itself, and the other one is one.

Those are the two numbers that it is divisible by.

And that's why I didn't want to say exactly

two other natural numbers, because one of the numbers

is itself.

And if this doesn't make sense for you,

let's just do some examples here,

and let's figure out if some numbers are prime or not.

So let's start with the smallest natural number-- the number 1.

So you might say, look, 1 is divisible by 1

and it is divisible by itself.

You might say, hey, 1 is a prime number.

But remember, part of our definition--

it needs to be divisible by exactly two natural numbers.

1 is divisible by only one natural number-- only by 1.

So 1, although it might be a little counter intuitive

is not prime.

Let's move on to 2.

So 2 is divisible by 1 and by 2 and not

by any other natural numbers.

So it seems to meet our constraint.

It's divisible by exactly two natural numbers-- itself,

that's 2 right there, and 1.

So 2 is prime.

And I'll circle the prime numbers.

I'll circle them.

Well actually, let me do it in a different color,

since I already used that color for the-- I'll

just circle them.

I'll circle the numbers that are prime.

And 2 is interesting because it is

the only even number that is prime.

If you think about it, any other even number

is also going to be divisible by 2, above

and beyond 1 and itself.

So it won't be prime.

We'll think about that more in future videos.

Let's try out 3.

Well, 3 is definitely divisible by 1 and 3.

And it's really not divisible by anything in between.

It's not divisible by 2, so 3 is also a prime number.

Let's try 4.

I'll switch to another color here.

Let's try 4.

Well, 4 is definitely divisible by 1 and 4.

But it's also divisible by 2.

2 times 2 is 4.

It's also divisible by 2.

So it's divisible by three natural numbers-- 1, 2, and 4.

So it does not meet our constraints for being prime.

Let's try out 5.

So 5 is definitely divisible by 1.

It's not divisible by 2.

It's not divisible by 3.

It's not exactly divisible by 4.

You could divide them into it, but you would get a remainder.

But it is exactly divisible by 5, obviously.

So once again, it's divisible by exactly two natural numbers--

1 and 5.

So, once again, 5 is prime.

Let's keep going, just so that we

see if there's any kind of a pattern here.

And then maybe I'll try a really hard one

that tends to trip people up.

So let's try the number.

6.

It is divisible by 1.

It is divisible by 2.

It is divisible by 3.

Not 4 or 5, but it is divisible by 6.

So it has four natural number factors.

I guess you could say it that way.

And so it does not have exactly two numbers

that it is divisible by.

It has four, so it is not prime.

Let's move on to 7.

7 is divisible by 1, not 2, not 3, not 4, not 5, not 6.

But it's also divisible by 7.

So 7 is prime.

I think you get the general idea here.

How many natural numbers-- numbers

like 1, 2, 3, 4, 5, the numbers that you learned when you were

two years old, not including 0, not including negative numbers,

not including fractions and irrational numbers and decimals

and all the rest, just regular counting positive numbers.

If you have only two of them, if you're only

divisible by yourself and one, then you are prime.

And the way I think about it-- if we

don't think about the special case of 1,

prime numbers are kind of these building blocks of numbers.

You can't break them down anymore

they're almost like the atoms-- if you think about what

an atom is, or what people thought

atoms were when they first-- they

thought it was kind of the thing that you couldn't divide

anymore.

We now know that you could divide atoms

and, actually, if you do, you might

create a nuclear explosion.

But it's the same idea behind prime numbers.

In theory-- and in prime numbers, it's not theory,

we know you can't break them down

into products of smaller natural numbers.

Things like 6-- you could say, hey, 6 is 2 times 3.

You can break it down.

And notice we can break it down as a product of prime numbers.

We've kind of broken it down into its parts.

7, you can't break it down anymore.

All you can say is that 7 is equal to 1 times 7,

and in that case, you really haven't broken it down much.

You just have the 7 there again.

6 you can actually break it down.

4 you can actually break it down as 2 times 2.

Now with that out of the way, let's think about some larger

numbers, and think about whether those larger numbers are prime.

So let's try 16.

So clearly, any number is divisible by 1 and itself.

Any number, any natural number you put up here

is going to be divisible by 1 and 16.

So you're always going to start with 2.

So if you can find anything else that goes into this,

then you know you're not prime.

And 16, you could have 2 times 8, you could have 4 times 4.

So it's got a ton of factors here

above and beyond just the 1 and 16.

So 16 is not prime.

What about 17?

1 and 17 will definitely go into 17.

2 doesn't go into 17.

3 doesn't go.

4, 5, 6, 7, 8, 9 10, 11-- none of those numbers,

nothing between 1 and 17 goes into 17.

So 17 is prime.

And now I'll give you a hard one.

This one can trick a lot of people.

What about 51?

Is 51 prime?

And if you're interested, maybe you

could pause the video here and try

to figure out for yourself if 51 is a prime number.

If you can find anything other than 1 or 51

that is divisible into 51.

It seems like, wow, this is kind of a strange number.

You might be tempted to think it's prime.

But I'm now going to give you the answer-- it is not prime,

because it is also divisible by 3 and 17.

3 times 17 is 51.

So hopefully that gives you a good idea

of what prime numbers are all about.

And hopefully we can give you some practice

on that in future videos or maybe some of our exercises.