 So another problem we can take a look at is divisions with remainders. So here's a problem to consider. Suppose I have 15 divided by 6. Now, if we try our first definition of division, what we're looking for is a whole number where the product 6 times something is equal to 15. And so the question is 6 times what is equal to 15? And the problem is we can't find such a number. 6 times 2 is equal to 12, so that doesn't work. 6 times 3 is equal to 18, and that doesn't work. And we might guess that the correct answer is somewhere between 2 and 3, but we don't have a whole number that's in between those two numbers. So there's two ways we can approach this. The first way we'll consider requires us to introduce an entirely new definition of division, or at least modify our existing definition. And this is the definition of division with remainder. And so this is a much more complex definition. This is let A, B, Q, and R be whole numbers. And we're going to make the requirement that R is between 0 and B. Could be equal to 0, but it has to be strictly less than B. And if my number A is the product BQ plus R, then I can say that A divided by B is Q with remainder R, and conversely. We say Q is the quotient, R is the remainder, and they're all related by this rather complicated arithmetic expression that combines multiplication, addition, and eventually division, and incidentally a subtraction there. All right, so let's start off with a basic problem. So let's prove 15 divided by 6 is equal to 2 with remainder 3. And so in order to approach this, this is approved. So we want to go back to our definition of division with remainder. And so we'll set down our definition for reference, and we'll compare what we want to prove with what our definition is. So let's take a look at this. This A must be 15, so I'll make that substitution. This B is 6, so I'll make that substitution. The quotient 2 is Q, so I'll make that substitution. And then my remainder 3 is R, and I'll make that substitution as well. So substituting the terms of what I want to prove into my definition, I get the following statement, and let's clean that up so it makes a little bit of a grammatical sense, and maybe I'll say something like since 0 is between 3 and 6, and 15 is 2 by 6 plus 3, then 15 divided by 6 is equal to 2 with the remainder of 3. And again, because multiplication and addition are part of my definition of division with remainder, I don't really have to go into a lot of detail about knowing why that's true. I can simply claim that it's true. Ideally, we actually wanted to be true, so let's make sure that that's the case. Let's see, 2 times 6 is 12, plus 3 is 15, 3 is, in fact, greater than or equal to 0 less than 6, so we're good there. That statement is true, so it follows 15 divided by 6 is 2 with the remainder of 3, and again, the essential part of our proof is the section that we're showing in green.