Intro to remainders

Let's take the number 7 and divide it by 3. And I'm going to conceptualize dividing by 3 as let me see how many groups of 3 I can make out of the 7. So let me draw 7 things-- 1, 2, 3, 4, 5, 6, 7. So let me try to create groups of 3. So I can definitely create one group of 3 right over here. I can definitely create another group of 3. So I'm able to create two groups of 3. And then I can't create any more full groups of 3. I have essentially this thing right over here left over. So this right over here, I have this thing remaining. This right over here is my remainder after creating as many groups of 3 as I can. And so when you see something like this, people will often say 7 divided by 3. Well, I can create two groups of 3. But it doesn't divide evenly, or 3 doesn't divide evenly into 7. I end up with something left over. I have a leftover. I have a remainder of 1. So this is literally saying 7 divided by 3 is 2 remainder 1. And that makes sense. 2 times 3 is 6. So it doesn't get you all the way to 7. But then if you have your extra remainder, 6 plus that 1 remainder gets you all the way to 7. Let's do another one. Let's imagine 15 divided by 4. Let me draw 15 objects-- 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15. Now, let me try to divide it into groups of 4. So let's see, that's one group of 4. That's another group of 4. And then that's another group of 4. So I'm able to create three groups of 4. But then I can't create a fourth full group of 4. I am then left with this remainder right over here. I have a remainder right over here of 3. I have 3 left over. So we could say that 15 divided by 4 is 3 remainder 3. 4 goes into 15 three times. But that only gets us to 12. 4 times 3 is 12. To get all the way to 15, we need to use our remainder. We have to get 3 more. So 15 divided by 4, I have 3 left over. Now, let's try to think about this doing a little bit of our long division techniques. So let's say that I have 4. Let's say I want to divide 75 by 4. Well, traditional long division techniques. 4 goes into 7 one time. And If you're looking at place value, we're really saying the 4 is going into 70 ten times, because we're putting this in the tens place. And then we say, 1 times 4 is 4. But really, once again, since it's in the tens place, this is really representing a 40. But either way, we subtract 4 from the 7. We get a 3. And then we bring down this 5. And we say 4 goes into 35. Well, let's see. 4 times 8 is 32. 4 times 9 is 36. That's too big. So it goes 8 times. 8 times 4 is 32. You subtract 35 minus 32 is 3. And 4 doesn't go into 3 anymore. So here I have this 3 left over. I have a remainder of 3. So you could say that 75 divided by 4 is equal to 18 remainder 3.