 So let's begin our discussion of multiplication by considering how you would multiply in base n. So, for example, let's take a typical multiplication, find the product 3 times 254 base 8. Now, here's where it's worth starting with the definition. This product 3 times 254 base 8 is the sum of 3 254s. So that's 254 base 8 plus 254 base 8 plus 254 base 8. And so I can do this as an addition because I know how to add in base 8. So we'll set up our addition using our place value table. And again, as long as I stay within the columns, I can, so to speak, add normally. So I'll add these numbers in the first column 2 plus 2 plus 2. It's going to give me 6, 5 plus 5 plus 5 is going to give me something I can write down as 15. 4 plus 4 plus 4 will give me something I'll write down as 12. And so far so good. Except for one minor problem. We're working in base 8. So this number 12, this number 15, they don't actually exist in base 8. So I need to do some bundling trading and combining. So we'll start with the 12 here, and I'm going to split that into a set of 8 and 4. And then I can trade the 8 for the 1 in the next place, and I'll combine these two to give me 16. Once again, I'm working in base 8, so I'll split this 16 into 8s, and then I'll trade this 8 for 1 and combine. And again, I have another 8 here I can trade, so I'll trade this 8 for 1 and combine. And one last step, I have an 8 here that I can trade over to the next place. Now I do have these two empty places here, so I've got to fill them in with 0s to indicate that they are places in my number. And my final step, I can record the digits and give the final answer by indicating that the sum, which is also equal to the product, is 1004 base 8. Now, this is a somewhat long-winded way of approaching the problem, but we can take a couple of shortcuts. So again, same problem, find 3 multiplied by 254 base 8, but again, I'm going to do this as a repeated addition, but let's consider what that repeated addition looks like. So again, we set it up this way. And the thing that's worth noting here is that within each column, I have a repeated addition. So if I take a look at this last column, I have 4 plus 4 plus 4. Well, that's the same as 3 times 4, so I'll just write it that way. Again, 5 plus 5 plus 5, that's 3 5s added together. And then 2 plus 2 plus 2, that's 3 2s added together. And so what I did before, well, I could have just started with 3 4s, 3 5s, and 3 2s. And so that suggests another way we can approach the problem is just to multiply each of the digits in our number by 3. So let's take a look at how we might do that for a different problem. So this is 7 times 257 base 8. And again, what this is is it's the sum of 7 257s. And I could write down a fairly lengthy addition problem and then do all of the different sums. But let's see if we can apply our shortcut. So rather than adding together 257, 257, 257, and so on, what I would have in this first column is I'd have 7 7s added together. I'd have 7 5s added together. And I'd have 7 2s added together. So this would be my first step. And now I could just do the multiplication normally. Again, as long as I stay within each column, I can be a little bit loose with the notation. So 7 times 2, I know what that is. 7 times 5, I know what that is. 7 times 7, I know what that is. And I can express, again, our product this way as long as I stay within each column. However, I'm not actually done with the problem yet. Because, again, these numbers have no meaning in base 8. So I'm going to bundle, trade, and combine in order to write down a number that is in base 8. So here I'll take the advantage of the fact that if I were to split this 49 into 48 and 1, that's a whole bunch of 8s altogether. 6 8s, so I'll trade. And then combine 35 and 6 gives me a 41. And again, I could split this into a 40 and 1. So there's 5 8s, and then 1 left over. And I'll trade and combine 14 and 5 gets me 19. And again, I'll split this into a 16 and 3. That's 2 8s, so I'll trade. And there's nothing to combine it with. And so that allows me to write my final product. 2 3 1 1, written in base 8.