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Multiples

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Multiples. The idea of multiples is actually pretty straightforward. The issue is that it's incredibly difficult and kind of frustrating to try to define what a multiple is in words, but if you understand the idea of multiplication, then you understand the idea of multiples. The biggest issue is that people tend to confuse multiples with factors.

So before we talk about any verbal definition, here's a nice helpful visual definition of how multiples and factors are different. So let's take the number 12 for instance, right Can you name a factor of 12. Well how about 6 and 2? If I'm doing a factor tree here. Right and breaking it down to its prime factors and then I can break 6 down further to 3 and 2.

And there we go I've reached the end of my factor tree. So 3,2,6,1 and 12 are all. All factors of 12 multiples just go in the other direction. So 12 times two is 24. 12 times three is 36. 12 times 4 is 48.

So the way to keep your factors and multiple straight is remember that factors are below the line. This line that I've just drawn right here. Those are factors and above the line are multiples. And as a mnemonic, multiply multiples get bigger. So how many multiples does a number have?

Well, an infinite number of multiples. I can keep multiplying 12 until the cows come home. But let's talk about an actual verbal definition. Now, just to just to preface this, the reason it's hard to put your finger on the verbal definition of a multiple is because it involves two numbers, right? So I could say something like this.

I could say if x is a multiple of why Then y is a factor of x. It's just the reverse. So, for instance, 36 is a multiple of 12. 12 is a factor of 36. You can also put it this way. If x is a multiple of y, Then x is divisible by y.

For instance, 36 is a multiple of 12 and 36 is divisible by 12. Finally, you could put it this way, a multiple of a number can be obtained by And multiply that number by an integer. So, take 12 and then multiply it by 2 you have 24. Great. 24 is a multiple of 12.

And again, my recommendation is don't worry about memorizing these verbal definitions. Just put it into that chart that we were just looking at. Okay, let's talk about the relationships between multiples of a certain number. If I add two multiples of a number together, the sum will itself be a multiple of that number.

So for instance, 24 is a multiple of 12 right? 36 is also a multiple of 12, we have it right on the screen here, 36 and 24. And summed together they make 60, is 60 a multiple of 12? It sure it. So if I add two multiples together, the sum of those two multiples will itself be a multiple of that number If I subtract two multiples of a number from each other, then the difference will itself be a multiple of that number.

So for instance, we take 48 which is a multiple of 12, right? And so is 24. And if I subtract 48-24, I get 24, which is a multiple of 12. And then finally if I multiply two multiples of a number together The product will itself be a multiple of that number.

So for instance, if I take the numbers 24 and 36, both of which are multiples of 12, I get 864. And you can check this on your calculator, is that a multiple of 12? Well, is it divisible by 12? Then yes, it is. One note, this does not work for division.

You can only do this for addition, subtraction and multiplication. I'm just gonna write that in, not division. Let's do some practice on this. If 16x=y, where x and y are integers Y must be a multiple of which of the following. And this is select all that apply.

So there could be more than one correct answer here. Go ahead and put your video on pause, try this yourself and then we'll talk about it. All right, so let's out with this premise. If 16x=y, then I can say that y is a multiple of 16. Right?

That's basically the definition of what multiples are. 16 times something equals y. So y is a multiple of 16. Unfortunately, I don't see 16 in this list here, so I can't pick it. But what I can do is break 16 down a little bit more. I could say for instance 2 times 8 times x equals y.

Well 2 times this thing whatever 8x equals is y. Well, that fits my multiple definition perfectly, right? y is now a multiple of 2. So A is one of the correct answers. And then I can just flip this around and say, well, couldn't I write 8 times 2 times x = y?

Well sure and then eight times this on none quantity equals y so y is a multiple of 8. Can I go any further? Yeah let's break eight down four times. Two times two times x equals y. So four times this whole unknown quantity right here equal to y therefore y is a multiple of four.

But I can't go any further than that unfortunately cuz if I break four down it's only gonna give me more twos and so the correct answers for this one are two 4 and 8. All right, a more difficult problem here. If K (K+ 200), and (K + 350) are multiples of P, then P could equal which of the following?

You know the drill, go ahead, put the video on pause and then we'll talk about it together. Here's what I know, I know that some number k, a bigger number k + 200, and then an even bigger number than that are all multiples of the smaller number P. So if I wanted to, I could represent this in the same way that I'd been doing before in this Chart right here, right?

P is the smallest number, down here at the bottom. And P times something gets me K, that's what a multiple is. P times another something, right, a different something, a bigger something, gives me K + 200. And then P times the biggest of these three gives me K + 350. What this problem is testing is your understanding of the multiple relationships.

So if you need to go back and review that a few minutes ago in this video, feel free to do that. But remember if I have two numbers that are both multiples of the same smaller number, Then the difference between those multiples the subtraction between those multiples is also a multiple that number. So let's take these two right here I got K + 200 and K What's the difference between those two?

Well, it's 200, right? So what does that mean? It means that 200 must be a multiple of P. Because K+200 is a multiple of P, K is a multiple of P. So the difference between them must also be a multiple of P again. Go back a few minutes if you need to refresh yourself on these multiple relationships.

Okay, so 200 must be a multiple of P P. What else? Well, what's the distance between K+ 350 and K+ 200? It's a distance of 150. Right? I went 150 more to get there.

So that means also since K + 350 is a multiple of P. And K + 200 is a multiple P. That 150 must be a multiple of P. So whatever number I pick here, has to be a factor of both of these two numbers. Or put differently, both of these two numbers have to be a multiple of whatever number I pick.

So let's run each of these numbers. Okay, is 200 a multiple of 20? Yeah, it is. Is 150 and multiple of 20?. No, I don't think so. So that's out.

Is 200 a multiple of 25? Well, 200 divided by 25 gives me eight. So yes, and is 150 and multiple is 25. Yeah, it is. Look at that. So that one works for both of them. Let's double check the other ones.

Is 200 a multiple of 75? Nope. It's not, and feel free to divide that in your calculator, if you wanna double-check. Is 150 a multiple of 75? Well, it is, but it doesn't matter, because 200's not, right?

How about 100, is 200 a multiple of 100? Yes, is 150, nope, doesn't work. And finally, 150, well, 150 is a multiple of 150. But 200 is not a multiple of 150. And so our correct answer here has to be B.

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