 Last time we computed these two products. So this time we're going to walk backwards through those. We're going to see what happens when you divide 413 by 13, for example. We expect we should get 25 out, but we'll see how that works. So I'll divide 413 by 13. First thing I can notice is that 13 is not going to divide into 4 in any base. This has two digits. This only has one. 13 is clearly larger than 4. So I can put a zero in that position. But 13 should divide into 41, and I'd like to know how many times. If I just looked up at my answer, I'd know it's 2, but I'd like to have some idea why it's 2. So let's consider what happens if I have 2 and 3. So if it's 2, then I get 2 times 13. So that would be 2 times 3, which is 10, plus 2 times 10, which is 20. So that would give me 30. 30 is less than 41, so 13 divides into 41, at least 2 times. But what happens if I do 3 times 13 instead? 3 times 3 is 13, plus 3 times 10, which is 30. That would give me 43, but 43 is larger than 41, so 13 does not divide into 41 at least 3 times. So I'll put the 2 here, and I saw that 2 times 13 was 30. So I'll subtract 30. That will leave me with 11, and then I'd bring down the 3. Now I need to know how many times 13 goes into 113. I can look at my table over here, and if this works out nicely, I should be able to add value in the 3's row to the same value in the 10's row, and get those to be equal to 113. So I can see 3 plus 10 is less than 113. 10 plus 20 is less than 113. 30 plus 13 gives me 43. That's less than 113. 40 plus 20 is 100, so that's less than 113. 50 plus 23 will give me 113. So that's exactly what I want. So that's from the 5's column. So if I have 5 times 13, that gives me 23 plus 50, which is 113. And I'll have a remainder of 0 here. So 13 times 25 gives me 413, which is exactly what I had before. Next, we'll try doing this for a larger problem. Unfortunately, this time I'm not just going to be able to look over my table and add a couple of terms together to figure out what some of these products are. It's going to end up being a lot more complicated, and I'm going to do a whole lot more multiplication to see what some of the products of one of these terms are. So I'm going to start with our product, and I'll divide it by, say, 3124. Now, again, we know it should go in 2,535 times, but we're going to pretend like we don't know that. So 3124 has four digits in it. It's not going to divide into anything smaller than another four-digit number. So I won't have to consider doing any real division until I get at least four digits on the right-hand side. On the other hand, this four-digit number starts with a 1, this one starts with a 3. So 3124 is clearly larger than 1330. I'll have to pull in a fifth digit. But a five-digit number is clearly going to be larger than a four-digit number. So I know 3124 will divide into 13305. I just have to figure out how many times it does. So I'm going to make a guess based on the largest digit here, which is the 3. And that I want the results of the multiplication to be smaller than this. So 3 times something should give me 13. If I pick 3, then this might work as 3 times 3 is 13. But that will be dependent on what happens in the rest of this. And I'm guessing 124 times 3 is going to be larger than 3 of 5. Specifically, once we get to this third digit, we're going to have problems. So I'm going to go with 2 times 3124. But since I don't know how much that is exactly, I'm going to go find out. So 3124 times 2. 2 times 4 is 12. 2 times 2 is 4. Plus 1 is 5. 2 times 1 is 2. And 2 times 3 is 10. So that looks reasonably close to our 13305. 5 minus 2 is 3. I want to do 0 minus 5, so I'll need to borrow something from the 3. Now I have 10 minus 5 is 1. 2 minus 2 is 0. 3 minus 0 is 3. So I did come out with something smaller than my 3124. So I've done my arithmetic correctly so far. Pulled out another 5. Now I need to multiply 3124 by something to get something slightly smaller than 3,000. I know if I multiply 3124 by 10, I would just get 3124. I'm going to try multiplying 3124 by 1 less than my base, which will be 5 in this case. So 5 times 4 will give me 32. 5 times 2 is 14. Plus 3 is 21. 5 times 1 is 5. Plus 2 is 11. 5 times 3 is 23. Plus 1 is 24. So that's reasonably close to our 30,000. As close as we're going to be able to get since if we multiplied by 10, then we should have incremented that 2 to a 3 instead. So 5 minus 2 is 3. 3 minus 1 is 2. 0 is 2. And I'll have a 10. So 10 minus 4 is 2. And 2 minus 2 is 0. So 2023 is less than 3124. And I multiplied by 5. Next I'll pull down the 1. So I've got a 5-digit number again, which will obviously be larger than my 4-digit number. So I'd like to find something to multiply my 3 by so that it's pretty close to 20. I could multiply 3 by 4. But I think once I multiply my 1 by 4, I'm going to get something that's too large. So I'm going to try multiplying this by 3. 4 times 3 will give me 20. 3 times 2 is 10 plus 2 is 12. 3 times 1 is 3. Plus 1 is 4. And 3 times 3 is 13. So this seems reasonably close to 20,000. 1, 1, borrow something. So this would be a 10. Minus 1 will give me 5. Now I'll have 12 minus 4, which is 4. 2 times 4 gives me 12. So 12 minus 4 is 4. 5 minus 3 is 2. 1 minus 1 is 0. So again, I've got something smaller than my 3124. So my arithmetic is going well. I multiplied by 3. So I'll write down a 3 there. Lastly, I'll bring down the 2. Now I want to find something to multiply 3124 by to get 24.112. So 5 seems like a decent choice. So I've already done 5. 5 times 3124 is indeed 24.112. So I can do that. That leaves me with 0 left over. So I've gone through the same process that we normally do for division. I'm just doing this in base 6. And I'm doing a whole lot of referring to my times table. Every time I needed to know what one of these factors were, I needed to go multiply my number by some single-digit number. So I ended up doing a lot of multiplication just to go over here and be able to do division.