 Welcome to this first video in a series on the division algorithm. This is a mathematical result that's actually really well known to everyone, although maybe not in the form that we're going to see. So think of a pair of integers for the time being making both of them positive. We know that the first integer won't always divide the second one evenly, but we do know that if we divide the second one by the first, it will go in a certain number of times with the remainder. For example, look at say 8 and 345. Now I'm pretty sure that 8 doesn't divide 345 just by looking at it, because 8 is even and 345 isn't. But whatever the case, I know that 8 goes into 345 a certain number of times. That number of times is called the quotient with a remainder leftover. We can find out the values of the quotient and the remainder by dusting off the old school process of long division. I won't talk through the details here, but just carry out the long division while you watch and listen. The important thing to notice here is that every time we write a multiple on top and then carry through the multiplication and then the subtraction, we get a remainder leftover. But that remainder may be so large that we can keep continuing the division process. Take that 25 there the first time we do the multiplication. Why don't we just stop there and say that 8 divides 345 a certain number of times with remainder 25? Well, in some ways, that's not incorrect. I could take 8 into 345 40 times and have 25 left over. That is, 345 is 40 times 8, that's 320, plus another 25, and that's correct. But the remainder here is so large that I know because of the process that I can divide again. And so we keep dividing until ultimately we can say that 8 goes into 345, 43 times with a remainder of 1. And I can write that expression as follows, 345 equals 8 times 43 plus 1. Note the quotient here is being multiplied to the divisor 8 and the remainder 1 is being added to the product to get that 345. It's this final addition slash multiplication step that's important. Notice what this does is allow me to talk about the result of long division without mentioning any division. I can see the quotient and the remainder at a glance. So let's see how well you understand this with a concept check. Consider the equation 2012 equals 55 times 36 plus 32. And that is a true equation, you can check that. But what does it mean? Here are four statements about what divides what, how many times with a remainder and I want you to select all that apply and don't cheat by actually using long division. Just see if you can tell merely by looking at the equation. So the answer here is both A and B. When we look at the equation 2012 equals 55 times 36 plus 32, we can pick off the divisor as one of the two things being multiplied. The quotient as the other and the remainder is the stuff left over that's being added back in. So this could be taken in one of two ways. Either 55 is the divisor and 36 is the quotient or vice versa. But either way you go, the remainder is 32. Just to check this, here's the long division process fully worked out for both cases. So the long division process lets us phrase division results without using any division. Can we always do this for any two integers? The answer is basically yes, and that is the result known as the division algorithm. So the division algorithm states that if we take any two integers A and B, with B bigger than zero, then there exist unique integers Q and R such that A is equal to B times Q plus R. And zero is less than or equal to R is less than B. So this is a lot of information here and to interpret this in light of what we know about division. The division algorithm says that if we divide B into A and B is a positive not zero integer. Then Q is the unique maximum number of times B goes into A and R is the remainder. B is the divisor here like eight was in the example earlier. Q is the quotient, that's the number of times B goes into A. And since we know that the remainder could be as small as zero, but can never be larger than B, because if it were, B could go into A more times. That's why the inequality is here. So let's see how well you're understanding the statement of the division algorithm with an example. In the statement of the division algorithm, let's suppose that A is 101 and B is 16. What are the values of Q and R that the division algorithm guarantees in this case? And here are your choices, and take a look at it and pause the video. So the right answer here is A. And we can check this by simply doing the math in the division algorithm. 101 is equal to 16 times 6, 16 times 6 is 96 plus 5 left over. But you might have noticed in trying to check these things that answers C and D also check out, 101 is also equal to 16 times 5 plus 21. And 101 is also equal to 16 times 7 plus negative 11. So Y is A right, but C and D are wrong. Well, it has to do with the inequality that's in the division algorithm, which states that the values of Q and R have to be, in this case, such that 0 is less than or equal to R, and R is less than 16. R has to be less than the thing I'm dividing by. Of the three values of Q and R in the concept check that make the equation work, only the one that's in A fits the inequality. The division algorithm also works if you divide into a negative number as well, which is a little different from long division, because we don't often use negative numbers in long division. So for an example, take A equal to negative 87 in the division algorithm and B equal to 6. So the division algorithm says that the divisor needs to be positive, but the number we're dividing into doesn't actually have to be. So we should be able to find a Q and an R such that negative 87 is equal to 6Q plus R. Now we can try some trial and error to see if we can find that Q. Multiplying by negative 15 gives us a nice round number of negative 90, which is actually pretty close to negative 87. Is that the right quotient? Will Q equal negative 90 work? Well, it depends on how the remainder works out. If we use Q equals negative 90 and get a remainder between 0 and 5, which is the range provided by the inequality in the division algorithm, then that Q value will work. So let's put in Q equals negative 90, and see what we have. Or sorry, Q equals negative 15, and see what we have. So negative 87 is equal to negative 15 times 6, that's the negative 90. And I'd have to add a 3 on to make that equation work. And so that's my remainder. So yes, this quotient value works. We can write negative 87 equal to 6 times negative 15 plus 3. So that's an overview of the division algorithm. Thanks for watching.