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Welcome to the greatest common divisor or greatest common factor video. So just to be clear, first of all, when someone asks you whether what's the greatest common divisor of 12 and 8? Or they ask you what's the greatest common factor of 12 and 8? That's a c right there for common. I don't know why it came out like that. They're asking you the same thing. I mean, really a divisor is just a number that can divide into something, and a factor-- well, I think, that's also a number that can divide into something. So a divisor and a factor are kind of the same thing. So with that out of the way, let's figure out, what is the greatest common divisor or the greatest common factor of 12 and 8? Well, what we do is, it's pretty straightforward. First we just figure out the factors of each of the numbers. So first let's write all of the factors out of the number 12. Well, 1 is a factor, 2 goes into 12. 3 goes into 12. 4 goes into 12. 5 does not to go into 12. 6 goes into 12 because 2 times 6. And then, 12 goes into 12 of course. 1 times 12. So that's the factors of 12. Let's write the factors of 8. Well, 1 goes into 8. 2 goes into 8. 3 does not go into 8. 4 does go into 8. And then the last factor, pairing up with the 1 is 8. So now we've written all the factors of 12 and 8. So let's figure out what the common factors of 12 and 8 are. Well, they both have the common factor of 1. And that's really not so special. Pretty much every whole number or every integer has the common factor of 1. They both share the common factor 2 and they both share the common factor 4. So we're not just interested in finding a common factor, we're interested in finding the greatest common factor. So all the common factors are 1, 2 and 4. And what's the greatest of them? Well, that's pretty easy. It's 4. So the greatest common factor of 12 and 8 is 4. Let me write that down just for emphasis. Greatest common factor of 12 and 8 equals 4. And of course, we could have just as easily had said, the greatest common divisor of 12 and 8 equals 4. Sometimes it does things a little funny. Let's do another problem. What is the greatest common divisor of 25 and 20? Well, let's do it the same way. The factors of 25? Well, it's 1. 2 doesn't go into it. 3 doesn't go into it. 4 doesn't go into it. 5 does. It's actually 5 times 5. And then 25. It's interesting that this only has 3 factors. I'll leave you to think about why this number only has 3 factors and other numbers tend to have an even number of factors. And then now we do the factors of 20. Factors of 20 are 1, 2, 4, 5, 10, and 20. And if we just look at this by inspection we see, well, they both share 1, but that's nothing special. But they both have the common factor of? You got it-- 5. So the greatest common divisor or greatest common factor of 25 and 20- well, that equals 5. Let's do another problem. What is the greatest common factor of 5 and 12? Well, factors of 5? Pretty easy. 1 and 5. That's because it's a prime number. It has no factors other than 1 and itself. Then the factors of 12? 12 has a lot of factors. It's 1, 2, 3, 4, 6, and 12. So it really looks like only common factor they share is 1. So that was, I guess, in some ways kind of disappointing. So the greatest common factor of 5 and 12 is 1. And I'll throw out some terminology here for you. When two numbers have a greatest common factor of only 1, they're called relatively prime. And that kind of makes sense because a prime number is something that only has 1 and itself as a factor. And two relatively prime numbers are numbers that only have 1 as their greatest common factor. Hope I didn't confuse you. Let's do another problem. Let's do the greatest common divisor of 6 and 12. I know 12's coming up a lot. I'll try to be more creative when I think of my numbers. Well, the greatest common divisor of 6 and 12? Well, it's the factors of 6. Are 1, 2, 3, and 6. Factors of 12: 1, 2, 3-- we should have these memorized by now. 3, 4, 6, and 12. Well, it turns out 1 is a common factor of both. 2 is also a common factor of both. 3 is a common factor of both. And 6 is a common factor of both. And of course, what's the greatest common factor? Well, it's 6. And that's interesting. So in this situation the greatest common divisor-- and I apologize that I keep switching between divisor and factor. The mathematics community should settle on one of the two. The greatest common divisor of 6 and 12 equals 6. So it actually equals one of the numbers. And that makes a lot of sense because 6 actually is divisible into 12. Well, that's it for now. Hopefully you're ready to do the greatest common divisor or factor problems. I think I might make another module in the near future that'll give you more example problems.

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Welcome to the greatest common divisor or greatest common factor video. So just to be clear, first of all, when someone asks you whether what's the greatest common divisor of 12 and 8? Or they ask you what's the greatest common factor of 12 and 8? That's a c right there for common.

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What is the least common multiple, abbreviated as LCM, of 15, 6, and 10? So the least common multiple is exactly what the word is saying. It's the least common multiple of these numbers. And I know that probably didn't help you much. But let's actually work through this problem. So to do that, let's just think about the different multiples of the 15, 6, and 10 and then find at the smallest multiple, the least multiple, they have in common. So let's find the multiples of 15. So you have 1 times 15 is 15. 2 times 15 is 30. Then if you add 15 again, you get 45. You add 15 again, you get 60. You add 15 again, you get 75. You add 15 again, you get 90. You add 15 again, you get 105. And if still none of these are common multiples with these guys over here, then we might have to go further. But I'll stop there for now. So that's the multiples of 15 up through 105. Obviously, we can keep going from there. Now, let's do the multiples of 6. 1 times 6 is 6. 2 times 6 is 12. 3 times 6 is 18. 4 times 6 is 24. 5 times 6 is 30. 6 times 6 is 36. 7 times 6 is 42. 8 times 6 is 48. 9 times 6 is 54. 10 times 6 is 60. 60 already looks interesting, because it is a common multiple of both 15 and 60, although we have two of them over here. We have a 30, and we have a 30. We have a 60 and a 60. So the smallest common multiple, so if we only cared about the least common multiple of 15 and 6, we would say it's 30. So let me write this down as an intermediate. The LCM of 15 and 6, so the least common multiple, the smallest multiple that they have in common, we see over here. 15 times 2 is 30. And 6 times 5 is 30. So this is definitely a common multiple. And it's the smallest of all of their common multiples. 60 is also a common multiple. But it's a bigger one. This is the least common multiple. So this is 30. Well, we haven't thought about the 10 yet. So let's bring the 10 in there. And I think you already see where this is going. Let's do the multiples of 10. They are 10, 20, 30, 40. Well, we already went far enough, because we already got to 30. And 30 is a common multiple of 15 and 6. And it's the smallest common multiple of all of them. So it's actually the fact that the LCM of 15, 6, and 10 is equal to 30. Now, this is one way to find the least common multiple. Literally just look at the multiples of each of the number and then see what the smallest multiple they have is in common. Another way to do that is to look at the prime factorization of each of these numbers. And the least common multiple is the number that has all of the elements of the prime factorizations of these and nothing else. So let me show you what I mean by that. So you could do it this way. Or you could say 15 is the same thing as 3 times 5. And that's it. That's its prime factorization. 15 is 3 times 5. Both 3 and 5 are prime. We can say that 6 is the same thing as 2 times 3. That's it. That's its prime factorization. Both 2 and 3 are prime. And then we can say that 10 is the same thing as 2 times 5. Both 2 and 5 are prime. So we're done factoring it. And so the least common multiple of 15, 6, and 10 just needs to have all of these prime factors. And what I mean, to be clear, is in order to be divisible by 15, it has to have at least one 3 and one 5 in its prime factorization. So it has to have at least one 3 and at least one 5. By having a 3 times 5 in its prime factorization, that ensures that this number is divisible by 15. To be divisible by 6, it has to have at least one 2 and one 3. So it has to have at least one 2. And we already have a 3 over here. So that's all we want. We just need one 3. So one 2 and one 3, this 2 times 3, ensures that we are divisible by 6. And let me make it clear. This right here is the 15. And then to make sure that we're divisible by 10, we have to have at least one 2 and one 5. These two over here make sure that we are divisible by 10. And so we have all of them. This 2 times 3 times 5 has all of the prime factors of either 10, 6, or 15. So it is the least common multiple. And so if you multiply this out, you will get 2 times 3 is 6. 6 times 5 is 30. So either way, hopefully, both of these resonate with you. And you see why they make sense. This second way is a little bit better if you're trying to do it for really complex numbers, numbers where you might have to be multiplying it for a long time. Either way, both of these are valid ways of finding the least common multiple.

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Least common multiple of three numbers (video) | Khan Academy

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What is the least common multiple, abbreviated as LCM, of 15, 6, and 10? So the least common multiple is exactly what the word is saying. It's the least common multiple of these numbers. And I know that probably didn't help you much. But let's actually work through this problem.

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William and Luis are in different physics classes at Santa Rita. Luis's teacher always gives exams with 30 questions on them, while William's teacher gives more frequent exams with only 24 questions. Luis's teacher also assigns three projects per year. Even though the two classes have to take a different number of exams, their teachers have told them that both classes-- let me underline-- both classes will get the same total number of exam questions each year. What is the minimum number of exam questions William's or Luis's class can expect to get in a given year? So let's think about what's happening. So if we think about Luis's teacher who gives 30 questions per test, so after the first test, he would have done 30 questions. So this is 0 right over here. Then after the second test, he would have done 60. Then after the third test, he would have done 90. And after the fourth test, he would have done 120. And after the fifth test, if there is a fifth test, he would do-- so this is if they have that many tests-- he would get to 150 total questions. And we could keep going on and on looking at all the multiples of 30. So this is probably a hint of what we're thinking about. We're looking at multiples of the numbers. We want the minimum multiples or the least multiple. So that's with Luis. Well what's going on with William? Will William's teacher, after the first test, they're going to get to 24 questions. Then they're going to get to 48 after the second test. Then they're going to get to 72 after the third test. Then they're going to get to 96. I'm just taking multiples of 24. They're going to get to 96 after the fourth test. And then after the fifth test, they're going to get to 120. And if there's a sixth test, then they would get to 144. And we could keep going on and on in there. But let's see what they're asking us. What is the minimum number of exam questions William's or Luis's class can expect to get in a year? Well the minimum number is the point at which they've gotten the same number of exam questions, despite the fact that the tests had a different number of items. And you see the point at which they have the same number is at 120. This happens at 120. They both could have exactly 120 questions even though Luis's teacher is giving 30 at a time and even though William's teacher is giving 24 at a time. And so the answer is 120. And notice, they had a different number of exams. Luis had one, two, three, four exams while William would have to have one, two, three, four, five exams. But that gets them both to 120 total questions. Now thinking of it in terms of some of the math notation or the least common multiple notation we've seen before, this is really asking us what is the least common multiple of 30 and 24. And that least common multiple is equal to 120. Now there's other ways that you can find the least common multiple other than just looking at the multiples like this. You could look at it through prime factorization. 30 is 2 times 15, which is 3 times 5. So we could say that 30 is equal to 2 times 3 times 5. And 24-- that's a different color than that blue-- 24 is equal to 2 times 12. 12 is equal to 2 times 6. 6 is equal to 2 times 3. So 24 is equal to 2 times 2 times 2 times 3. So another way to come up with the least common multiple, if we didn't even do this exercise up here, says, look, the number has to be divisible by both 30 and 24. If it's going to be divisible by 30, it's going to have to have 2 times 3 times 5 in its prime factorization. That is essentially 30. So this makes it divisible by 30. And say, well in order to be divisible by 24, its prime factorization is going to need 3 twos and a 3. Well we already have 1 three. And we already have 1 two, so we just need 2 more twos. So 2 times 2. So this makes it-- let me scroll up a little bit-- this right over here makes it divisible by 24. And so this is essentially the prime factorization of the least common multiple of 30 and 24. You take any one of these numbers away, you are no longer going to be divisible by one of these two numbers. If you take a two away, you're not going to be divisible by 24 anymore. If you take a two or a three away. If you take a three or a five away, you're not going to be divisible by 30 anymore. And so if you were to multiply all these out, this is 2 times 2 times 2 is 8 times 3 is 24 times 5 is 120. Now let's do one more of these. Umama just bought one package of 21 binders. Let me write that number down. 21 binders. She also bought a package of 30 pencils. She wants to use all of the binders and pencils to create identical sets of office supplies for her classmates. What is the greatest number of identical sets Umama can make using all the supplies? So the fact that we're talking about greatest is clue that it's probably going to be dealing with greatest common divisors. And it's also dealing with dividing these things. We want to divide these both into the greatest number of identical sets. So there's a couple of ways we could think about it. Let's think about what the greatest common divisor of both these numbers are. Or I could even say the greatest common factor. The greatest common divisor of 21 and 30. So what's the largest number that divides into both of them? So we could go with the prime factor. We could list all of their normal factors and see what is the greatest common one. Or we could look at the prime factorization. So let's just do the prime factorization method. So 21 is the same thing as 3 times 7. These are both prime numbers. 30 is, let's see, it's 3-- actually, I could write it this way-- it is 2 times 15. We already did it actually just now. And 15 is 3 times 5. So what's the largest number of prime numbers that are common to both factorizations? Well you only have a three right over here. Then you don't have a three times anything else. So this is just going to be equal to 3. So this is essentially telling us, look, we can divide both of these numbers into 3 and that will give us the largest number of identical sets. So just to be clear of what we're doing. So we've answered the question is 3, but just to visualize it for this question, let's actually draw 21 binders. So let's say the 21 binders so 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21. And then 30 pencils, so I'll just do those in green. So 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Let me just copy and paste that. This is getting tedious. So copy and paste. So that's 20 and then paste that is 30. Now, we figured out that 3 is the largest number that divides into both of these evenly. So I can divide both of these into groups of 3. So for the binders, I could do it into three groups of 7. And then for the pencils, I could do it into three groups of 10. So if there are three people that are coming into this classroom, I could give them each seven binders and 10 pencils. But that's the greatest number of identical sets Umama can make. I would have three sets. Each set would have seven binders and 10 pencils. And we essentially are just thinking about what's the number that we can divide both of these sets into evenly, the largest number that we can divide both of these sets into evenly.

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William and Luis are in different physics classes at Santa Rita. Luis's teacher always gives exams with 30 questions on them, while William's teacher gives more frequent exams with only 24 questions. Luis's teacher also assigns three projects per year.

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Become familiar with common fractions and decimals and converting between them.

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Worked example: Order of operations (PEMDAS)

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Worked example: Order of operations (PEMDAS)

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Now that we've got the basics of order of operations out of the way, let's try to tackle a really hairy and beastly problem. So here, we have all sorts of parentheses and numbers flying around. But in any of these order of operations problems, you really just have to take a deep breath and remember, we're going to do parentheses first. Parentheses. P for parentheses. Then exponents. Don't worry if you don't know what exponents are, because this has no exponents in them. Then you're going to do multiplication and division. They're at the same level. Then you do addition and subtraction. So some people remember PEMDAS. But if you remember PEMDAS, remember multiplication, division, same level. Addition and subtraction, also at the same level. So let's figure what the order of operations say that this should evaluate to. So the first thing we're going to do is our parentheses. And we have a lot of parentheses here. We have this expression in parentheses right there, and then even within that we have these parentheses. So our order of operations say, look, do your parentheses first, but in order to evaluate this outer parentheses-- this orange thing-- we're going to have to evaluate this thing in yellow right there. So let's evaluate this whole thing. So how can we simplify it? Well, if we look at just inside of it, the first thing we want to do is simplify the parentheses inside the parentheses. So you see this 5 minus 2 right there? We're going to do that first no matter what. And that's easy to evaluate. 5 minus 2 is 3. And so this simplifies to-- I'll do it step by step. Once you get the hang of it, you can do multiple steps at once. So this is going to be 7 plus 3 times the 5 minus 2, which is 3. And all of those have parentheses around it. And of course, you have all the stuff on either side-- the divide 4-- no. Oops. That's not what I want. I wanted to copy and paste. I want to copy and paste that right there. So copy, then-- no, that's giving me the wrong thing. It would've been easier-- let me just rewrite it. That's the easiest thing. I'm having technical difficulties. So divided by 4 times 2. And on this side, you had that 7 times 2 plus this thing in orange parentheses there. Now, at any step you just look again. We always want to do parentheses first. Well, you keep wanting to do and is there really no parentheses left? So we have to evaluate this parentheses in orange here. So we have to evaluate this thing first. But in order to evaluate this thing, we have to look inside of it. And when you look inside of it, you have 7 plus 3 times 3. So if you just had 7 plus 3 times 3, how would you evaluate it? Well, look back to your order of operations. We're inside the parentheses here, so inside of it there are no longer any parentheses. So the next thing we should do is-- there are no exponents. There is multiplication. So we do that before we do any addition or subtraction. So we want to do the 3 times 3 before we add the 7. So this is going to be 7 plus-- and the 3 times 3 we want to do first. We want to do the multiplication first. 7 plus 9. That's going to be in the orange parentheses. And then you have the 7 times 2 plus that, on the left hand side. You have the divided by 4 times 2 on the right hand side. And now this-- the thing in parentheses-- because we still want to do the parentheses first. Pretty easy to evaluate. What's 7 plus 9? 7 plus 9 is 16. And so everything we have simplifies to 7 times 2 plus 16 divided by 4 times 2. Now we don't have any parentheses left, so we don't have to worry about the P in PEMDAS. We have no E, no exponents in this. So then we go straight to multiplication and division. We have a multiplication-- we have some multiplication going on there. We have some division going on here, and a multiplication there. So we should do these next, before we do this addition right there. So we could do this multiplication. We could do that multiplication. 7 times 2 is 14. We're going to wait to do that addition. And then here we have a 16 divided by 4 times 2. That gets priority of the addition, so we're going to do that before we do the addition. But how do we evaluate that? Do we do the division first, or the multiplication first? And remember, I told you in the last video, when you have 2-- when you have multiple operations of the same level-- in this case, division and multiplication-- they're at the same level. You're safest going left to right. Or you should go left to right. So you do 16 divided by 4 is 4. So this thing right here-- simplify 16 divided by 4 times 2. It simplifies to 4 times 2. That's this thing in green right there. And then we're going to want to do the multiplication next. So this is going to simplify to-- because multiplication takes priority over addition-- this simplifies to 8. And so you get 14-- this 14 right here-- plus 8. And what's 14 plus 8? That is 22. That is equal to 22. And we are done.

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Worked example: Order of operations (PEMDAS) (video) | Khan Academy

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The order of operations is a rule that tells the correct sequence of steps for evaluating a math expression. We can remember the order using PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

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The order of operations tells us the order to solve steps in expressions with more than one operation. First, we solve any operations inside of parentheses or brackets. Second, we solve any exponents. Third, we solve all multiplication and division from left to right. Fourth, we solve all addition and subtraction from left to right.

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Learn to compare expressions with positive and negative fractions.

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Let's have some practice adding and subtracting negative numbers. So the first example I want to look at is 2 minus 3. So right now I'm just subtracting a positive number from another positive number, but you might already see that I'm subtracting a larger number from a smaller number. So I'm probably, or I will, definitely end up with a negative number. But let's just think about this a little bit. And I'm going to do it with a number line. So there's my number line right over there. Now this is 0, this is 1, this is 2, this is negative 1, this is negative 2. We could view this as starting at 2. So this is 2 right over here, and then we're going to subtract 3 from that 2. So we're going to move 3 to the left on the number line. So we're going to move 3 to the left, 1, 2, 3. And that gets us to negative 1. This is equal to negative 1. Now let's mix it up a little bit more. Let's imagine what would happen if we had negative 2 minus 3. So this was positive 2 minus 3. Now let's think about negative 2 minus 3. So once again, let's draw our number line. And I'll put 0 over here. So this is 0, this is 1, this is negative 1, negative 2, negative 3, negative 4, negative 5, negative 6, and I could keep going. But we're starting at negative 2, and then we're subtracting 3 again. So once again, we're going to move three to the left of negative 2. So we go 1, 2, 3. We end up at negative 5. So this is negative 5. So notice in both situations we subtracted 3, we moved 3 to the left on the number line. It's just here we started 2 to the right of 0. Here we started 2 to the left of 0. This is negative 2. Let's do another example with these same numbers. Let's imagine negative 2 plus 3. I encourage you to pause this video, and try to think about this on your own. So we could draw the number line-- I could draw a straighter number line than that-- so draw the number line again. And let's say that this is negative 2, negative 1, 0, 1, and 2 again. We're starting at negative 2, we're starting 2 to the left of zero. So we're starting at negative 2, and we're going to add 3. So we're going to go 3 to the right now, 1, 2, 3, and we end up at positive 1. Now let's think about 2, so positive 2, and we're going to subtract a negative 3. And other videos we've already talked about this. In fact, there's a video explaining why this actually makes sense. But when you subtract a negative, this is the same thing as adding the positive. So 2 minus negative 3 is the exact same thing as 2 plus 2 plus positive 3. These two statements are equivalent, and this just boils down to, this right over here, is just going to be 5. Now, let's mix it up a little bit more. Let's imagine negative 2 minus negative 3. Now, this might seem really intimidating to have all of these negatives in place here, but you just have to remember subtracting a negative, like this, is going to get you a positive. So this is the exact same thing as negative 2 plus 3, and negative 2 plus 3, we've already seen it right over here. You start at negative 2, you start 2 to the left of 0, and then we're going to go 3 to the right, we're adding 3, 1, 2, 3.

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Adding & subtracting negative numbers (video) | Khan Academy

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Learn how to add and subtract negative numbers. The problems solved in this video are 2 - 3 = -1 and -2 - 3 = -5 and -2 + 3 = 1 and 2 - (-3) = 5 and -2 - (-3).

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Numbers and Operations | 8th grade | Khan Academy

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Math is all about numbers (like 89 and 3.14) and operations (like addition and multiplication). In these tutorials, we learn about a some new types of numbers and some new types of operations.

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How to work with Positive and Negative Numbers

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How to work with Positive and Negative Numbers

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Expressing a number in standard form or scientific form

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Expressing a number in standard form or scientific form

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Expressing a number in standard form or scientific form

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Standard form or scientific notation is used to express long numbers in a more compressed form as this video shows.

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Whole Numbers - Reasoning Strategies - About the Assessment - Math Reasoning Inventory

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This section describes the specific reasoning strategies and understandings that relate to whole numbers. Reasoning strategies are organized into two categories-those appropriate and those not appropriate to the numbers at hand in the problem.