Prime Numbers

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.