 Hi there, and welcome to the screencast on integer divisibility. So we're gonna, in the screencast, go way back to your elementary school days, actually, and pick up a basic concept, when does one whole number divide another? And make it a little more formal so we can do some more interesting math with it. So here's an arithmetic question for you, and you can pause the video once you have read the question if you want to. We'll come back to the answer when you're done. Which of these numbers, these are all whole numbers or integers, is divisible by seven? So you've got five of these numbers to think about. Think about it and pause the video and come back when you think you have an answer. Okay, so we're back, I'm assuming you have an answer here, and hopefully this is pretty easy to figure out here. Well, 84 is certainly divisible by seven. Now, how do you know that 84 is divisible by seven? One sort of naive way is to pull out your calculator and take 84 and literally divide it by seven and see that it goes in evenly, that you get 12 out. That's okay, but let's try to be technology independent if we can and to see why is it that we know that 84 is divisible by seven. This will pay off for us a little bit later. We can go back to long division, if you remember that, and take seven and divide it into 84 and just carry out the long division. So seven goes into eight once, put down the seven, subtract, get 14, divide seven into 14, it goes twice. That's a two, I have a 14 here, and there's no remainder. And that's what we typically mean by seven divides 84 is that it divides evenly, there's no remainder. Now, keep in mind too, and this is really important here that the bookkeeping we've done up here on the top can be rewritten. This really means if seven goes into 84 12 times with no remainder, what that could be rewritten to say is that 84 is equal to 12 times seven. So another way to say that 84 is divisible by seven is to say that 84 is a multiple of seven. Those two phrases are really the same thing. Now let's see if we can use that idea to apply to the other numbers here. Now 14, or negative 14, of course, is also divisible by seven. Now why is that? We don't typically think of having negative numbers in long division. And again, naively, we could just pull out a calculator, but can we rewrite it like this? Can we take negative 14 and write it as something times seven? Is a negative 14 a multiple, an integer multiple, I should say, of seven? And of course it is, because I could fill in that blank with negative two. So 84 and negative 14 are both divisible by seven for the same reason. I can set up this little equation here and write 84 or negative 14 as an integer multiple of seven. Now what about 76? Naively, again, the calculator way of answering this would tell us no way. That's not going to be divisible by seven. If I punch in 76 divided by seven, I get a decimal answer. Now what that means in terms of our alternative way of thinking about it is if I set up 76 equals blank times seven, could I put an integer into this blank to make the equation true? And the answer is going to be no. There's no integer that goes here that will make this equation work. So 76 is not divisible by seven. 3 million, 64, 877 is also not divisible by the same reason. If you carry out the long division on this, it's long division. It takes quite a struggle, not really a struggle, just a lot of grunge work to get through here. I don't know how many times it goes, blah, blah, blah, blah, blah. And I believe that is the remainder here you get, you do get a remainder and it's four. So the remainder is not zero, since that's not divisible by seven. What about the number zero, is that divisible by seven? That's a little hard to think about too with long division somehow, unless we use this alternative way of looking at it. So let's set up the equation is zero and integer multiple of seven. And I put an integer in this blank that makes the equation true. And of course I certainly can, I can put in the integer zero. So zero is actually divisible by seven. Zero is divisible by seven as well, because it's a multiple of seven. So we're thinking about divisibility in terms of multiplication, which is kind of a nice way to think about divisibility. Because division is actually quite a difficult operation to understand. So we can understand whether an integer is divisible by another integer by seeing if it's a multiple of that integer. And that is what leads us to our big definition in this section. We're going to say that a non-zero, check that, non-zero, we never divide by zero here. A non-zero integer m divides an integer n, provided that there is an integer q such that n equals m times q. What this is really saying is that m divides n, if n is a multiple of m, get your n's and m straight. What this is really saying here is that n is an integer multiple of m. Integer multiple of m, just like 84 was an integer multiple of seven. Therefore, seven divides 84. We also say in some different ways that m is a divisor of n, and that m is a factor of n. And then we use this notation m divides n. This is not a fraction here, this is not the fraction m over n. We're actually staying really, really clear of fractions here. We don't do fractions at this point in the course. So this is just a sentence that says m divides n. Let's look at some examples here of how we instantiate this definition. So two divides ten, and that's how you would read this. Two divides ten because ten is an integer multiple of two. Namely, there is an integer, namely five, that I can put here. That's the q in the definition. Ten is equal to five times two. There is an integer that I can multiply two by to get ten. Similarly, negative three divides 27 because 27 is an integer multiple of negative three. I'm sorry, negative nine times negative three. I can write an integer in this blank that makes this equation true. Seven divides zero because zero is equal to zero times seven. All three of these examples, I'm putting in an integer into that blank to make the equation true. Zero does not divide anything. That's excluded explicitly in the definition. Division by zero is undefined. It doesn't equal anything, not infinity or something crazy like that. Finally, five does not divide 12. That's how you would read that symbol. Because there's no integer q such that 12 is equal to five q. So now it's time for a concept check. Which of the following five statements is or are true? And just pause the video and select all that apply. Okay, now that we're back, let's just tick down the list here. This is not true. This says 14 divides seven. 14 divides seven, and that's not true. The other way around would be true, seven divides 14. If 14 divided seven, then I would have to be able to fill in the following blank with an integer, okay? Now I could put a fraction in this blank, namely the fraction one half and make this true, but there's no integer that goes here that makes that true. So 14 does not divide seven. However, seven does divide 14. This is a one way kind of operation. Eight does divide negative eight, why? Because I can take negative eight and write it as some integer times eight. Namely, I could put in the integer negative one into that blank. So this is good. Ten does divide zero. We've seen a couple of examples of that already because I could take this equation, set it up, and put an integer in this blank, namely zero. This is not correct because zero does not divide anything. Does 12 divide 782? So we can determine whether 12 divides 782. One way to do this is by writing out multiples of 12. What are those multiples of 12? If I start with the positive multiples of 12, I'd have 12, 24, 36. Let's make a list of these guys here, 48. Let's skip a bit and we will eventually, after a long time, after 60 of these end up at 720, that's a 12 times 60. And let's keep going from there. I would have 732, 744. And what I want to do is see if 782 shows up in the list of multiples of 12. This is slightly an inefficient way to do this, but it will become helpful for us in another section where we talk about integer congruence coming up, 780, and then the next multiple is 792. And so I have skipped right over 782. It's not a multiple of 12. It's a multiple of 12 plus two, so it's got a bit of a remainder there. So 12 does not divide 782 because 782 is not a multiple of 12. So we've learned quite a formal definition of divisibility here. Dephrasing divisibility in terms that have nothing to do with division, which actually turns out to be pretty handy for us. Thanks for watching.