Remainders, this is an important topic. So far in discussing division, we've talked mostly about multiples and their factors. When we divide any integer by one of its divisors, that quotient is always an integer, and in fact that integer is always another one of the number's factors.

So for example, 21 divided by 3 is 7. Well, 7 of course is a factor of 21. In fact, this is related to the fact that 21 divided by 7 = 3. Both of them are connected to the fact that the product of 3 and 7 is 21, those are both factors of 21. What happens when we divide one number by another number that is NOT one of the factors, one of the divisors of the first number?

For example, what happens if we divide 17 by 5? And we talked about this a little bit in a previous video we're gonna review here. One approach would be to change the fractions. So a perfectly valid answer to that question, what is 17 divided by 5? A perfectly valid answer is the improper fraction 17 over 5. That number is 17 divided by 5.

Sometimes we need to change that to a mixed numeral. Here we can write this as a mixed numeral, 3 and two-fifths. And of course, what that mixed numeral means is that there's implicit addition. What that really means is that it's 3 + two-fifths. This is a perfectly mathematically correct answer, and in some context, this is exactly what's required.

Sometimes the test will ask you for some kind of division and the numbers will be listed as mixed numerals or as decimals, and so this is the way you have to go. Sometimes though, we wanna keep everything in terms of integers. For example, suppose we have 17 identical items, and we can form some kind of important set from groups of 5 of these items, so we're grouping identical items.

In that case, we'd wanna know how many full sets can we create and we'd want to know how many extra will be left over after we create those full sets. Here, clearly, we can form 3 full sets of 5, and then we would have 2 extra items left over. In other words, the quotient would be 3 and the remainder would be 2. In this video, I will use the term mixed-numeral quotient and integer quotient to distinguish between these two types of division.

So if we're talking about 17 divided by 5, the mixed-numeral quotient is 3 and two-fifths. So I could also write that mixed-numeral quotient as a decimal if I wanted. The integer quotient, I'm gonna state it this way verbally, 5 goes into 17 three times, with a remainder of 2. That three is the integer quotient.

So notice that I'm not writing this as an equation, I'm writing that in verbal form. Some of you might have seen in schools something along the lines of this, 17 divided by 5 = 3 remainder 2. That's a slick little notation. I'm not gonna use that only because, first of all, some people find that confusing, and second of all, the test never uses, that you will not see on the test.

The test always states this stuff in verbal form, so I'll be stating it in verbal form here. In these problems, find the integer quotient and the remainders. Try and do this without a calculator, pause the video, and then I'll talk about these. 20 divided by 6, well, obviously 6 goes into 18 three times, and we have 2 left over.

95 divided by 7, this is a little bit trickier. You have to know that 91 equals 7 times 13, and so it goes in 13 times with 4 left over. 56 divided by 8, of course, 8 goes into 56 an even number of times, it goes in 7 times, and we have no remainder. So for that, we can actually just write the equal sign and note that we have no remainder at all.

Let's talk about the terminology here, 20 divided by 6 yields 3, with a remainder of 2. What are the terms of all these rolls? The 20, the number divided, is called the dividend. The 6, the number by which we're dividing, is called the divisor. Now, be very careful here, we're using divisor in a slightly different sense than we've used in previous videos.

In previous videos, we were talking about divisor as a synonym for factor, the factors of a number, the divisors of a number. So we're not using divisor in its factor sense here, because, of course, 6 is not a factor of 20. We're just using divisor in the sense of the number by which we're dividing. So this is divisor in a division sense, a slightly different use of the word.

Of course, 3 is the integer quotient and 2 is the remainder. Well, a few things to notice, first of all, notice that the remainder is always less than the divisor. The remainder has to be a positive number, could be 0, could be positive, but it has to be less than the divisor. If it were greater than the divisor, the divisor could go into it one more time.

So that's why it's always less than the divisor. If D is the dividend, S is the divisor, Q is the quotient, and r is the remainder, then we can write things in this form. D divided by S, dividend divided by divisor, = the quotient, the integer quotient, + the fraction remainder over divisor. So this is an important formula because it allows us to make a connection between the integer quotient and the mixed-numeral quotient.

Q is the integer quotient, but if we add that little fraction, remainder over divisor, that gives us the mixed-numeral quotient. Now, it's very important to realize also, Q is an integer, so any non-integer part of the mixed numeral quotient is gonna be that fraction. This means that if the quotient is written in mixed numeral or decimal form, the non-integer part of the quotient is remainder over divisor.

That's a really subtle and important point. Suppose we're told we divide two numbers and we get that horrible long number with the decimal part. We know that the remainder divided by the divisor will equal that non-integer part, that decimal 0.375, which happens to equal the fraction three-eighths. We don't necessarily know that r=3 and S=8.

It could be that we have a remainder 3 in the divisor of 8, that could be true, but r over S could also equal any fraction equivalent to 3 over 8. So it could be 6 over 16, 9 over 24, 12 over 32, etc. So all we have here is ratio information. We know that the ratio of r over S equals 3 over 8. And if we knew either the remainder or the divisor, we could find the other.

One important skill is generating examples of possible dividends that, when divided by a certain divisor, yield a specific remainder. For example, what numbers when divided by 12 have a remainder of 5? Well, clearly, the simplest one would just be 5 + 12, which is 17. When we divide 17 by 12, it goes in once with a remainder of 5. Similarly, we could take any multiple of 12 and simply add 5.

So all of those are numbers that when we divide by 12, it will go in a certain number of times and then it will have a remainder of 5. And notice we can get from one to the next simply by adding or subtracting 12. So 17 plus 12 is 29, plus 12 is 41, plus 12 is 53. So much in the same way as we can get from one multiple to the next by adding the number, we can also get from one of these numbers to the next by adding the number.

If I tell you that 1997 is a number that, when you divide by 12, has a remainder 5, then you could find other numbers of this set by adding or subtracting 12. So we could add 12, and that would be 2009. We could subtract 12, that would be 1985. Incidentally, what we're doing here, we would be finding all the years that in the Chinese calendar were the Year of the Ox.

So I'm mentioning this only cuz it's a really unexpected application of remainders and divisors. There's all kinds of subtle ways that remainder and divisor questions can show up on the test. Another tricky question is, what is the smallest positive integer that, when divided by 12, has a remainder of 5?

Well, you may think that it's 17, but it's actually 5. Well, wait a second, what's going on here? 12 is bigger than 5, so if we divide 5 by 12, it goes into it zero times, an integer coefficient of zero, and the remainder is 5. In general, if the divisor is larger than the dividend, then the integer quotient is 0 and the remainder equals the dividend.

The test absolutely loves to test questions about this. Finally, a very important topic is rebuilding the dividend. What do I mean by this? Above, we have that fraction equation. If we multiply both sides of the equation by S, the divisor, we clear all the fractions and we get the equation dividend equals divisor times quotient plus remainder.

That formula is immensely important. That is called the rebuilding the dividend equation, and that is an immensely important formula for problem solving. Let me give an example. Here's a question, pause the video, and then I'll talk about a solution to this. When positive integer N is divided by positive integer P, the quotient is 18, with a remainder of 7.

When N is divided by (P + 2), the quotient is 15 and the remainder is 1. What is the value of N? So we are being asked to find the dividend. So using the rebuilding the dividend formula for the first sentence, we get N = 18 times P + 7. Using the formula for the second sentence, we get N = 15 times (P + 2) + 1, the remainder.

And of course, I can simplify this to 15P + 31. Those are two expressions for N, so I can set them equal to each other, Subtract 15P, subtract 7, we get 3P = 24, P = 8, and then we can plug in and we can solve for the dividend, which is 151. So using that formula is a very powerful problem-solving tool.

In summary, dividing by a number that is not a factor of the dividend, we can get either a mixed-numeral quotient or an integer-quotient and a remainder. The remainder is always greater than or equal to 0 and less than the divisor. r over S is the non-integer part of a mixed numeral or a decimal quotient. That's the remainder divided by the divisor. We can always set it equal to that non-integer part.

One important skill is generating examples of possible dividends for a given divisor and remainder. And finally, we talked about the very important rebuilding the dividend formula.