>> Hi there and welcome the Screencast on integer divisibility. So we're going to in this 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 cause 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 whole numbers or integers is divisible by 7? So we 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. OK, 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 7. Now how do you know that 84 is divisible by 7? One sort of naive way is to pull out your calculator and take 84 and literally divide it by 7 and see that it goes in evenly, that you get 12 out. That's OK but let's try to be technology independent if we can and just see why is it that we know that 84 is divisible by 7. This will pay off for us a little bit later. We can go back to long division, if you remember that, and take 7 and divide it into 84 and just carry out the long division. So 7 goes into 8 once. Put down the 7, subtract and get 14, divide 7 into 14 it goes twice. That's a 2. I have a 14 here and there's no remainder. And that's what we typically mean by 7 divides 84 is that it divides evenly. There's no remainder. Now keep in mind too, and this is really important here that this the bookkeeping we've done up here on the top can be rewritten. And this really means if 7 goes into 84 12 times in the remainder, what that could be rewritten to say is that 84 is equal to 12 times 7. So another way to say that 84 is divisible by 7 is to say that 84 is a multiple of 7. 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 7. 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 7? Is negative 14 a multiple, an integer multiple I should say of 7. And of course it is because I could fill in that blank with negative 2. So 84 and negative 14 are both divisible by 7 for the same reason. Again, set up this little equation here and write 84 or negative 14 as an integer multiple of 7. Now what about 76? Natively again the calculator way of answering this would tell us no way, that's not going to be divisible by 7. If I punch in 76 divided by 7 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 7, 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 7. 3,764,877 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 between blah, blah, blah, blah, blah. And I believe that is the remainder here you get, you do get a remainder and it's 4. So the remainder is not 0, hence that's not divisible by 7. What about the number 0? Is that divisible by 7? 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 0 and integer multiple of 7. Can I put an integer in this blank that makes the equation true? And of course I certainly can. I can put in the integer 0. So 0 is actually divisible by 7. 0 is divisible by 7 as well, because it's a multiple of 7. 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 in integer n, provided that there is an integer 2, such that n equals m times q. What this is really saying is that m divides n if n is a multiple m. Get your n's and m's 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 7, therefore a 7 divides 84. We also way 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 were actually staying really, really clear of fractions. We don't do fractions at this point in the course. So this is just a sentence, this says m divides n. Let's look at some examples here of how we instantiate this definition. So 2 divides 10 and that's how you would read this, 2 divides 10 because 10 is an integer multiple of 2. Namely there is an integer, namely 5 that I could put here. That's the q in the definition, 10 is equal to 5 times 2. There is an integer that I can multiple 2 by to get 10. Similarly negative 3 divides 27 because 27 is an integer multiple of negative 3 -- I'm sorry negative 9 times negative 3. I can write an integer in this blank that makes this equation true. 7 divides 0 because 0 is equal to 0 times 7. All three of these examples I'm putting in an integer into that blank to make the equation true. 0 does not divide anything. That's excluded explicitly in the definition. Division by zero is undefined. It's doesn't equal anything, not infinity or something crazy like that. Finally five does not divide 12. That's how we would read that symbol because there's no integer q's such that 12 is equal to 5q. 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. OK, now that we're back let's just tick down the list here. This is not true. This says 14 divides 7, 14 divides 7 and that's not true. The other way around would be true, 7 divides 14. If 14 divided 7 then I would have to be able to fill in the following blank with an integer. OK, now I could put a fraction in this blank, namely the fraction 1/2 and make this true, but there's no integer that goes here that makes that true. So 14 does not divide 7. However 7 does divide 14. So this is a one way kind of operation. 8 does divide negative 8, why because I can take a negative 8 and write it as some integer times 8. Namely I could put it in the integer negative 1 into that blank. So this is good. 10 does divide 0. We've seen a couple of examples of that all ready because I could take this equation, set it up and put an integer in this blank, namely 0. This is not correct because zero does not divide anything. Does 12 divide 782? Well, 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 2. 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. De-phrasing divisibility in terms that have nothing to do with division, which actually turns out to be pretty handy for us. Thanks for watching.