Prime numbers

in this video I want to talk a little bit about what it means what it means to be a prime number and what you'll see in this video or would 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 don't have to worry about it right now just need to know that prime numbers are pretty 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 natural number and a natural number once again just as an example these are like the numbers 1 2 3 so essentially the counting number starting at 1 or you could say the positive integers it is a natural number divisible by divisible by exactly exactly two two numbers or two other natural numbers actually I shouldn't say two other I should say two natural numbers so it's not necessarily it's not two other natural numbers divisible by exactly two natural numbers one of those numbers is itself itself and the other one is one those are the two numbers that 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 figure out if numbers are prime or not so let's start with the smallest natural number the number one so you might say look one is divisible by one and it is divisible by itself you might say hey one is a prime number but remember part of our definition it needs to be divisible by exactly two natural numbers one is divisible by only one natural number only by one so one although it might be a little counterintuitive is not one is not is not prime let's move on to two let's move on to two so two is divisible by one and by two and not by any other natural numbers so it seems to Mayor cut straight it has to it's divisible by exactly two natural numbers itself that's two right there and one so two is prime so two is prime and I'll circle the prime numbers I'll Circle them well actually let me do it in different colors and I already use that color for the well I'll just circle them I'll circle the numbers that are prime now let's three and it too 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 two above and beyond one and itself so it won't be prime well think about that more in future videos let's try out three well three is definitely divisible by one and three and it's really not divisible by anything in between it's not divisible by 2 so 3 is also it's also a prime number let's try 4 and 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 actually has so it's divisible by 3 natural numbers 1 2 & 4 so it does not meet our constraint for being prime let's try out let's try out 5 so 5 is definitely divisible by 1 it's not divisible by 2 it's not just a little by 3 it's not just divisible by 4 exactly difficult by 4 you would you could divide them into it but you would get a remainder but it is exactly divisible by 5 obviously so once again exactly it's divisible by exactly two natural numbers 1 & 5 so it 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 let's try the number let's try the number 6 it is divisible by 1 it is visible by two it is divisible by three not four or five but it is divisible by six 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 is divisible by it has four so it is not prime let's move on to seven seven is divisible by one not two not three not four not five not six but it's also divisible by seven so seven is prime I think you get the general idea here how many natural numbers numbers like one two three four five the numbers that you learned when you were when you were two years old not including zero not including negative numbers not including fractions and in irrational numbers and decimals and all the rest just regular counting positive numbers if you having 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 one prime numbers are kind of these building blocks of numbers that you can't break them down any more they're almost like the atoms you know 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 oh and in prime numbers it's not there we know you can't break them down into smaller into products of smaller natural numbers things like things like six 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're just you just have the 7 there again 6 you can actually break it down for 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 me try let's try 16 so clearly any number is divisible by 1 and itself so any number any natural number you put up here is going to be divisible by and 16 so you're always going to start with two so if you can find anything else that goes into this then you know you're not prime and 16 you could have two times eight you can have four times four so it's got a ton of factors here above and beyond just the one in 16 so 16 is not prime what about 17 1 go 1 in 17 will definitely go into 17 2 doesn't go into 17 3 doesn't go four five six seven eight nine ten eleven of those numbers nothing between nothing nothing between one and seventeen goes into 17 so 17 is 17 is prime and now I'll give you a hard one I'll give you a hard one this one can trick a lot of people what about 51 is 51 prime 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 like wow this is kind of a strange number you might be tempted to think its 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 we can we can give you some practice on that in future videos or maybe some of our exercises