Prime numbers | Prime time | Class 6 | Mathematics | Khan Academy

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[00:00:00]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 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.

[00:00:20]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. And 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 one 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 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 prime.

[00:02:15]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 meet our constraint. It's divisible by exactly two natural numbers itself. That's two right there and one. So two 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 use that color for the I'll just circle them. I'll circle the numbers that are prime. And two is interesting because it is the only even number that is prime.

[00:02:55]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. We'll think about that more in future videos. Let's try out three.

[00:03:09]Well, three is definitely divisible by 1 and three. And it's really not divisible by anything in between. It's not divisible by two. So, three is also a prime number. Let's try four. I'll switch to another color here. Let's try four. Well, four is definitely divisible by 1 and four, but um it's also divisible by two.

[00:03:31]2 * 2 is 4. It's also divisible by two.

[00:03:34]So, it's divisible by three natural numbers.

[00:03:38]1 2 and four. So it does not meet our constraints for being prime. Let's try out five. So five is definitely divisible by one. It's not divisible by two. It's not divisible by three. It's not exactly divisible by four. You could divide them into it, but you would get a remainder. But it is exactly divisible by five obviously. So once again it's divisible by exactly two natural numbers 1 and five. So once again five 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 six. It is divisible by one. It is divisible 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 it is divisible by. It has four. So it is not prime. Let's move on to seven. 7 is divisible by 1. Not 2, not 3, not four, not five, not six, but it's also divisible by seven. 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 2 years old. Not including zero. Not including negative numbers.

[00:05:13]Not including fractions and irrational numbers and decimals and all the rest.

[00:05:18]Just regular counting positive numbers.

[00:05:22]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 you don't think about the special case of one, 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 six, you could say, "Hey, 6 is 2 * 3."

[00:06:05]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 * 7. And in that case, you really haven't broken it down much. You just have the seven there. Again, six, you can actually break it down. four. You can actually break it down as 2 * 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 one 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 two. So, if you can find anything else that goes into this, then you know you're not prime. And 16, you could have 2 * 8, you could have 4 * 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. Two doesn't go into 17. Three 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 * 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.