Prime Numbers

Identifying prime numbers

In this video I want to talk a little bit about

what it means to be a prime number

and what you will hopefully see in this video

is this pretty straightforward concept

but as you progress through your mathematical career

you'll see that there is actually fairly sophisticated concepts

that can be built on top of the idea of the 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 that prime numbers are

pretty important. So I'll give you the definition

and the definition might be a little confusing

but when we see it with examples it should be pretty straightforward

A number is prime if it is a natural number

for example 1, 2 or 3 (the counting numbers starting at 1)

or you could also say "the positive integers"

it is a natural number divisible by exactly two natural numbers: itself and 1.

Those are the two numbers that it's divisible by.

If this does not make sense for you lets just do some examples.

Lets figure out if some numbers are prime or not.

So lets start with the smallest natural numbers.

The number 1. So you might say "1 is divisible by 1"

and "1 is divisible by itself", 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 only by one natural number, only by 1.

So 1, even it may be a little counter intuitive, is not prime.

Lets move on to 2.

So 2 is divisible by 1 and by 2, and not by any other natural numbers.

So it seems to fit our constraints.

It's divisible by exactly two natural numbers.

Itself and 1. So the number 2 is prime.

I will circle the numbers that are prime.

The number 2 is interesting because

it's the only even number that is prime.

If you think about it, any other even number

is also going to be divisible by 2., so it won't be prime.

We'll think about that more in future videos.

Lets try out 3. Well, 3 is definitely divisible by 1 and 3

and it's not divisible by anything in between.

it's not divisible by 2. So 3 is also a prime number.

Lets try 4.

4 is definitely divisible by 1 and 4, but

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.

Lets try out 5.

5 is definitely divisible by 1,

It's not divisible by 2, 3 or 4

(you could divide 5 / 4 but you would get a remainder)

And it is exactly divisible by 5, obviously.

So once again, 5 is divisible by exactly two natural numbers: 1 and 5

So once again, 5 is prime. Lets keep going,

so that we see if there is any kind of a pattern here

and then maybe I'll try a really hard one

that tends to trip people up. So lets try the number 6.

It is divisible by 1, 2, 3 and 6.

So it has four natural number "factors",

I guess you could say it that way

So it does not have exactly two numbers that it's divisible by,

it has four, so it is not prime.

Lets move on to 7.

7 is divisible by 1, not 2, 3, 4, 5 or 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 learn when you are two years old

not including zero, 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 are only divisible by yourself and by 1,

then you are prime.

and the way I think about it,

if we don't think of the special case of 1,

prime numbers are kind of these building blocks of numbers.

You can't break them down anymore.

They are almost like the atoms.

If you think about what the atom is,

or what people thought atoms were when they first...

they thought they were these things

you couldn't divide anymore.

We now know we could divide atoms and actually

if you do you may create a nuclear explosion.

But it's the same idea behind prime numbers. In theory, no prime number is not a theory.

You can't break them down

into products of smaller natural numbers.

Things like 6 you can 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 it's parts.

7 you can't break it down anymore.

All you can say is 7 equals 1 times 7.

And in that case you haven't really 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 lets think about

some larger numbers, and think about

whether those larger numbers are prime.

So lets try 16.

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

So 16 is divisible by 1 and 16.

So you are going to start with two,

so if you can find anything else that goes into this

then you know you are not prime.

And for 16 you could have 2 times 8,

you can have 4 times 4,

so it has 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, ...

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 are interested you can pause the video here

and try to figure out by 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 this 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 and maybe in some of our exercises.