Introduction to remainders

We're already somewhat familiar with the idea of division. If I were to say 8 divided by 2, you could think of that as 8 objects: 1, 2, 3, 4, 5, 6, 7, 8. Divided into equal groups of two. So how many equal groups of two could you have? Well, you could have one, two, three, or four groups of two. And so you'd say eight divided by two is equal to four.

Another way we could have thought about that is, you have one, two, three, four, five, six, seven, eight. If you were to divide it into two equal groups, well, you could have one group of four. Let me make it a little bit cleaner: one group of four, and then a second group of four. So two equal groups. How many in each of those equal groups? Well, there are four in each of those groups.

And so once again, eight divided by two is equal to four. Now we're going to extend our knowledge of division by starting to think about things that don't divide evenly. So what if we were to say, what is 8 divided by 3? Pause this video and see if you can think about that a little bit.

Alright, so let's draw eight objects again: 1, 2, 3, 4, 5, 6, 7, 8. One way to think about it, as we thought about it here, is can we divide this into groups that all have three in them? And how many groups would we be able to make of three? Let's try it out. I can make this group of three. I can make this second group of three, but I can't make any more groups of three.

And what I have left over are these two. The way that you would describe this—or one way to describe this is, "Hey, I was able to make two groups of three." So it's equal to two, and there's some left over. There's a remainder. Let me write that down: an important concept there is a remainder of 2 as well.

And so sometimes it's written as just a lowercase r: a remainder of 2. Another way to think about it is 2: this 2 times 3 is 6, and then if you were to put back that remainder, that's how you can get to 8.

Now, another way you could think about it is how we thought about in the second example with 8 divided by 2. Let me draw 8 objects again: 1, 2, 3, 4, 5, 6, 7, 8. You could say, "Hey, let me divide that 8 into 3 equal groups." So pause this video and see if you can divide this into three equal groups and then what might be left over.

Alright, so I'm going to try to divide this into three equal groups. I'm not going to be able to put four in each of those groups because I can only make two equal groups of four. I'm not gonna be able to put three into those three equal groups because that would actually be nine for doing that.

So each of my groups are going to have to be two. So I could make one group of two, another group of two, and there you go: three equal groups of two. I was able to sort out three equal groups of two with just 6. But once again, I have a remainder. I'm not able to make use of these two. They're not able to fit into one of, in this case, one of the three equal groups.

If I said four equal groups, then they would fit in. But if I just said 3 equal groups because I'm dividing by 3, then I have this left over. Again, let's do one more example. What if I were to ask you, what is 13 divided by 4? Pause this video and think about it, and as you might imagine, there will be a remainder involved.

Alright, well, let's draw 13 objects: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13. We could try to divide this into equal groups of four. That's one way to think about it. So let's see: that's a group of four. I have one group of four; that's a group of four. I have two groups of four, and then that is a group of four.

So I'm able to find three equal groups of four, so this is equal to three. Another way to think about it: four goes into thirteen three times, but then I have this little lonely circle here. I have one left over. I have a remainder of one because four times 3 gets you to 12, but then if you want to get to 13, well then you've got to throw in that remainder there.