 So how else could we perform a division? Well, one possibility is we can decompose the divisor. So this comes from our definition of division, as well as factoring. Suppose I have a product A times quantity BC is equal to D, then by the definition of division, A is D divided by this quantity BC. However, if I pull in some of the properties of multiplication, I can do some other things with this expression. So, for example, I have A is equal to C times B, and that's just commutativity. If I have a product B times C, I can reverse it C times B. And now I have a product of three terms, A, C, and B, and so associativity says I can group the first two terms if I feel like it, and I have AC times B equals D. Again, this is my associativity, and then my definition of division says that AC is D divided by B. But in fact, I can apply the definition of division a second time. If I divide by C, then I get A is D divided by B, then divided by C. Again, that's by my definition of division. And again, remember, equals means is the same as, and so what I have, A is the same as D divided by BC, and it's also the same as D divided by C, then divided by C. So what that tells me is that if I want to do D divided by BC, it's the same as A, which is the same as D divided by B divided by C. So I can do this division by doing this pair of divisions. And importantly, because our convention by the order of operations is that we perform divisions from left to right, I don't actually need to indicate this pair of parentheses here. I can just write it as D divided by B divided by C. And my order of operations says I'm going to do that divide by B first, then the divide by C. And so what we can do, if we want to, is we can divide by decomposing the divisor. For example, consider the division 612 divided by 36. Now, I could divide by 36 using the long division algorithm, but if I don't feel like doing that, or perhaps I haven't learned that long division algorithm yet, that's not a problem. I could break 36 into a product of two numbers. 36 is, among other things, 6 times 6. And so if I want to divide by 36, I can divide by 6 times 6. And by the preceding that says I can divide by 6, and then divide by 6. So I'll go ahead and do those divisions one at a time. 612 divided by 6, that's 102, divided by 6 is 17. Now, I don't actually have to do the division that way. If I want to break 36 up into other ways, I can do that as well. 36 is 4 times 9, for example, and so I can do the division by 36 as division by 4 times 9, divide by 4, then divide by 9. Again, importantly, we are doing our divisions from left to right, so we do our divide by 4 first, then we divide by 9, once again arriving at our quotient, 17. Again, one of the features of this is you get to pick how you're going to decompose the divisor. If you don't want to work with 4 and 9, maybe you could split that 36 up and do even smaller pieces. 36 is 6 times 6, and I'll break that 6 into 2 times 3. So 36 is 2 times 3 times 6. So if I wanted to do this division by 36, I could divide by 2, then by 3, then by 6. And again, the advantage here is that I have replaced a complicated division by 36 with a bunch of little easy divisions. So 612 divided by 2, well, that's something we should be able to do in our head, that's 306, and then now 306 divided by 3, again something we could probably do in our head, that's 102, divide by 6, maybe a little bit more difficult to do in our heads, but well, if we don't know that the answer is 17, we could also think about this as divide by, again, 2 divided by 3, because our 6 factors, 2 times 3. So 102 divided by 6, either you can go straight to the answer 17, or you can consider 102 divided by 2 is 51 divided by 3, 17, and there's our final answer.