 So let's take our definition of division apart. So suppose we're trying to find A divided by B, and remember that we can look at this in one of two ways. In a partitive division, we're going to take A and split it up into B sets. And there'll be some size that we don't know, and there'll be some things left over, and this leads to the following definition. If I have ABQ in our whole numbers, where the thing that's left over is less than the number of sets, then if A is B times Q plus R, then A divided by B is Q, with the remainder R, and conversely. Our original collection A has been broken up into B sets of size Q, and there's R things left over. The other possibility is I might be doing a quotitive division. So if I take A divided by B as a quotitive division, then I know that each set that I produce is of size B. I don't know how many sets that I produce, and then I'll have my leftovers. So again, we have this definition. If I have ABQ in our whole numbers, where R is less than B, then if A gets me Q sets each of size B plus my leftovers, then A divided by B is Q, with the remainder R, and importantly, conversely. But multiplication is commutative. The only difference between these two definitions is that I have BQ here and QB there. So because multiplication is commutative, these two definitions are really the same definition. Now we'll go ahead and name each part in our division statement. A divided by B is Q, with remainder R. We say A is the dividend, B is the divisor, Q is the quotient, and R is the remainder. So let's take a look at a problem, prove 11 divided by 4 equals 2, with remainder 3. Because this is a prove statement, we have to go back to the definition. So we'll go ahead and quote that. And I can just compare our definition and our statement. If I compare my statement to my definition, well, let's see. I have A must be 11, B must be 4, Q must be 2, R has to be 3, and so I'll fit them in. And I have 11A equals B times Q plus R. 11 is 4 times 2 plus 3, then 11 divided by 4 is 2, with remainder 3. And as per usual, because multiplication, addition, and this inequality are all part of the definition, we don't really have to say anything other than to claim this is true. Now of course, if it's not true that 11 is 4 times 2 plus 3, then our statement is also not going to be true as well. So you do want to make sure that the statement you put here is actually a true statement. So let's try a harder problem. Suppose N is 37,189 times 3,891 plus 17, and we want to find N divided by 3,891. Okay, well, here's where actually knowing what you know makes this more difficult, because you're tempted to do this problem the hardest way possible, which is to find this product and sum, and then to divide to get an answer. But the thing to remember is that at this point, we haven't actually introduced any method of doing this division. So if you want to put yourself in the position of a student, you don't know how to divide. So how are you going to solve the problem? Well, since we know the definitions, we can do mathematics. So let's go back to our definition. Our definition of divide tells us that if A equals Q times B plus R, where R is between 0 and B, then A divided by B is Q with remainder R. And since we know the definitions, we can do mathematics. And again, we'll compare our statement to our definition, and we see that A and N are the same. Q, 37189, B, 3891, R, 17. So if I want to find A divided by B, that's A divided by B, N divided by 3891, it must be the other number plus 17. And so that gives us our quotient and remainder. Well, on occasion, we may run into the following situation. Let's take a look at something like 27. Well, that's 3 times 9 plus 0. And I can say two things. If I view this partatively, I have three sets of nine apiece with nothing left over. So 27 divided by 3 gives you nine sets with nothing left over. Or I could look at this quotatively. So again, if I look at it quotatively, with 27, I have three sets of nine. So 27 divided into sets of nine give you three sets, and there's nothing left over. If the remainder is 0, we might be tempted not to say that the remainder exists, and we might simply write 27 divided by 3 equals 9 and not indicate a remainder. Or likewise, 27 divided by 9 equals 3, and again, not indicate a remainder. In these cases, we might say that 27 has been divided evenly, but how you speak influences how you think. And when we say even, well, even has a mathematical definition, the problem is there's nothing even about these numbers. There's nothing even about the numbers here. So saying this has been divided evenly suggests something that isn't true. And so it's probably better to say that there is no remainder. And even better, this implies the remainder doesn't exist. Well, it does actually exist. What exists is that there is a remainder of 0.