Prime Numbers Identifying prime numbers
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- 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.
- 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
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