Least Common Multiple
The Least Common Multiple. In this video, it's gonna be important to understand everything from the last video on the Greatest Common Factor. So if you haven't seen the previous video, I highly suggest that you watch that before watching this video. The Least Common Multiple is, as we'll see later in this video, identical to the Least Common Denominator.
LCM, LCD, I'll be using those two abbreviations, really two different abbreviations for exactly the same thing. Let's explore the Least Common Multiple with a simple example. Let's find the Least Common Multiple of 8 and 12. Well certainly, we can find the multiples of 8, those are the multiples of 8. We can find the multiples of 12, those are the multiples of 12.
Notice that there are some numbers that appear on both lists, they are the common multiples, the multiples that 8 and 12 have in common. We can separate those out and put them on a separate list, these are the common multiples, the multiples that both numbers have in common, that list of course is infinite. The lowest member is the least common multiple, which equals 24 here, 24 is the Least Common Multiple of 8 and 12.
Notice that the product of the two numbers is always one of the common multiples, but often not the Least Common Multiple of the two numbers. For example, the product of 8 and 12 was 96, that was on the common multiple list, but it was not the Least Common Multiple. In fact, the Least Common Multiple is much smaller than that product. Of course, we always can list the multiples to find the Least Common Multiple, but if the numbers are large, listing might be an impractical scenario.
We need a shortcut, especially since the test is absolutely not gonna ask you to find the Least Common Multiple of 8 and 12, it's gonna ask you about three-digit numbers, so we need some kind of shortcut here. Not surprisingly, the shortcut involves the prime factorization of the two numbers, as well as finding the Greatest Common Factor. So suppose we have to find the Least Common Multiple of 24 and 32, already a slightly more sophisticated problem.
Well, certainly, we can find the prime factorizations and the Greatest Common Factor. I assume that you can find, at this point, the prime factorization of the two numbers and the Greatest Common Factor, which here is 8. Now, write each number in the form, Greatest Common Factor times another factor, well, 24 is clearly 8 times 3, 32 is 8 times 4.
The Least Common Multiple is the product of those three factors, 8, and 3, and 4, of course that's 8 times 12, that's 96. That's the Least Common Multiple of 24 and 32. So that basic procedure, which I've demonstrated here, now I can demonstrate in a general form in a symbolic representation. Let M and N be the two numbers whose Least Common Multiple we want to find.
Let G be the Greatest Common Factor of these two numbers. Let N over G equal A, so that we're writing N as the Greatest Common Factor times another number, A. Let M over G equal B, so that we're writing M as the Greatest Common Factor times another number, B. Well then, the Least Common Multiple is just G times A times B.
In other words, the Greatest Common Factor times those leftover factors in each of the numbers. What is the Least Common Multiple of 12 and 75? I recommend that you pause the video here and work on this on your own, and then I'll give the solution. So of course the first thing we are gonna do is find the Greatest Common Factor, in fact, for this one, you shouldn't need to do any procedure, this is one with number sense.
You should just be able to look at it and see that the Greatest Common Factor is 3, that's the only factor that these two numbers have in common. Now, of course, 12 equals 3 times 4, 75 equals 3 times 25, those are the three factors we multiply together. The Least Common Multiple is 3 times 4 times 25, well, very convenient, 4 times 25 is 100.
So this just equals 300, and that is the Least Common Multiple. Why is it useful to find the Least Common Multiple? Well, for a few reasons, first of all, the test could ask you to find the least multiple directly, that could be a problem, find the Least Common Multiple of these two numbers. And sometimes word problems can ask for the Least Common Multiple in hidden ways.
So for example, something along the lines of, hot dogs are sold 12 to a pack, but hot dog buns are sold 8 to a pack. What is the least number of hot dogs and the least number of buns that you'd have to buy, so that you'd have a bun for every hot dog? That sort of thing, if you're matching things that come in sets of different sizes, often that is a Least Common Multiple question in disguise.
Also, the Least Common Multiple is, it's very important in adding and subtracting fractions because the Least Common Multiple is the Least Common Denominator. And of course, when you're adding and subtracting fractions, you have to find a common denominator. And especially if you're trying to do it quickly without a calculator, it is convenient to find the Least Common Denominator.
Here's a fraction problem, pause the video here, try and work this out on your own, and then I'll talk about the solution. So the first thing we're gonna do is find the Greatest Common Factor, and again, the Greatest Common Factor here, you should not need a procedure. Just with number sense, you should be able to see that the Greatest Common Factor is 5.
Clearly, 10 is 5 times 2, 35 is 5 times 7, multiply those three factors, and we get 70, that is the Least Common Multiple, which is the Least Common Denominator. 70 is the Least Common Denominator, so we wanna get each of those fractions a denominator of 70. So that means that the first one will multiply by 7 over 7, the second one we'll multiply by 2 over 2, we'll get 7 over 70 minus 2 over 70, that gives us 5 over 70, which we can simplify to 1 14th.
And that is the actual answer that would appear on the test. Some further tips on the Least Common Multiple. If A is a factor of R, then the Least Common Multiple of a number, and its multiple of a factor, and the number of which it is a factor must be R. So for example, the Least Common Multiple of 8 and 24 has to be 24, that's very important.
Another important thing, if it's obvious that A and B have no factors in common greater than 1, in other words, the Greatest Common Factor is 1, then the Least Common Multiple would have to be their product, A times B. So for example, the Least Common Multiple of 7 and 15 must be 7 times 15. And again, this is the type of thing you should just be able to see, you shouldn't have to follow any procedure to see that 7 and 15 have no factors in common.
Now, we leave it to this, I have outlined a procedure here, DO NOT, DO NOT, DO NOT rely on this procedure as your only method of finding the Least Common Multiple, you MUST develop your intuition. This is a really important point. If your only strategy for the integer properties is memorizing procedures, you are absolutely guaranteeing that you will have a mediocre performance in Math on the test.
To have an above-average performance, you must develop some number sense. I'm going to recommend an exercise that will help build number sense related to these skills. Everyday, pick a pair of two-digit numbers. Perhaps you could generate them randomly, pick them out of a hand, or get a calculator to generate them, perhaps you could get a friend to pick them randomly for you.
Get the pair of numbers, then in your head, without writing anything down, do your best to figure out the Greatest Common Factor and the Least Common Multiple. When you have your best guesses, then follow the whole procedure outlined here to find these values to check yourself. If you do this with a friend or study partner, you can quiz each other and check each others work.
And of course the point is to develop your mental Math abilities, develop your intuition, so that sometimes you'll just see the Greatest Common Factor, you won't have to rely on a procedure to get it. In summary, we talked about what the Least Common Multiple is and how to find it from listing factors. We talked about how to find the Least Common Multiple using Greatest Common Factor.
We pointed out that the Least Common Multiple is the Least Common Denominator, and this simplifies fraction addition and subtraction. We talked about some further tips for the Least Common Multiple. And we outlined an exercise for building number sense in intuition for the Greatest Common Factor and the Least Common Multiple. And once again, one of the very best things you can do for yourself on the math section is to develop your number sense.
Read full transcriptLCM, LCD, I'll be using those two abbreviations, really two different abbreviations for exactly the same thing. Let's explore the Least Common Multiple with a simple example. Let's find the Least Common Multiple of 8 and 12. Well certainly, we can find the multiples of 8, those are the multiples of 8. We can find the multiples of 12, those are the multiples of 12.
Notice that there are some numbers that appear on both lists, they are the common multiples, the multiples that 8 and 12 have in common. We can separate those out and put them on a separate list, these are the common multiples, the multiples that both numbers have in common, that list of course is infinite. The lowest member is the least common multiple, which equals 24 here, 24 is the Least Common Multiple of 8 and 12.
Notice that the product of the two numbers is always one of the common multiples, but often not the Least Common Multiple of the two numbers. For example, the product of 8 and 12 was 96, that was on the common multiple list, but it was not the Least Common Multiple. In fact, the Least Common Multiple is much smaller than that product. Of course, we always can list the multiples to find the Least Common Multiple, but if the numbers are large, listing might be an impractical scenario.
We need a shortcut, especially since the test is absolutely not gonna ask you to find the Least Common Multiple of 8 and 12, it's gonna ask you about three-digit numbers, so we need some kind of shortcut here. Not surprisingly, the shortcut involves the prime factorization of the two numbers, as well as finding the Greatest Common Factor. So suppose we have to find the Least Common Multiple of 24 and 32, already a slightly more sophisticated problem.
Well, certainly, we can find the prime factorizations and the Greatest Common Factor. I assume that you can find, at this point, the prime factorization of the two numbers and the Greatest Common Factor, which here is 8. Now, write each number in the form, Greatest Common Factor times another factor, well, 24 is clearly 8 times 3, 32 is 8 times 4.
The Least Common Multiple is the product of those three factors, 8, and 3, and 4, of course that's 8 times 12, that's 96. That's the Least Common Multiple of 24 and 32. So that basic procedure, which I've demonstrated here, now I can demonstrate in a general form in a symbolic representation. Let M and N be the two numbers whose Least Common Multiple we want to find.
Let G be the Greatest Common Factor of these two numbers. Let N over G equal A, so that we're writing N as the Greatest Common Factor times another number, A. Let M over G equal B, so that we're writing M as the Greatest Common Factor times another number, B. Well then, the Least Common Multiple is just G times A times B.
In other words, the Greatest Common Factor times those leftover factors in each of the numbers. What is the Least Common Multiple of 12 and 75? I recommend that you pause the video here and work on this on your own, and then I'll give the solution. So of course the first thing we are gonna do is find the Greatest Common Factor, in fact, for this one, you shouldn't need to do any procedure, this is one with number sense.
You should just be able to look at it and see that the Greatest Common Factor is 3, that's the only factor that these two numbers have in common. Now, of course, 12 equals 3 times 4, 75 equals 3 times 25, those are the three factors we multiply together. The Least Common Multiple is 3 times 4 times 25, well, very convenient, 4 times 25 is 100.
So this just equals 300, and that is the Least Common Multiple. Why is it useful to find the Least Common Multiple? Well, for a few reasons, first of all, the test could ask you to find the least multiple directly, that could be a problem, find the Least Common Multiple of these two numbers. And sometimes word problems can ask for the Least Common Multiple in hidden ways.
So for example, something along the lines of, hot dogs are sold 12 to a pack, but hot dog buns are sold 8 to a pack. What is the least number of hot dogs and the least number of buns that you'd have to buy, so that you'd have a bun for every hot dog? That sort of thing, if you're matching things that come in sets of different sizes, often that is a Least Common Multiple question in disguise.
Also, the Least Common Multiple is, it's very important in adding and subtracting fractions because the Least Common Multiple is the Least Common Denominator. And of course, when you're adding and subtracting fractions, you have to find a common denominator. And especially if you're trying to do it quickly without a calculator, it is convenient to find the Least Common Denominator.
Here's a fraction problem, pause the video here, try and work this out on your own, and then I'll talk about the solution. So the first thing we're gonna do is find the Greatest Common Factor, and again, the Greatest Common Factor here, you should not need a procedure. Just with number sense, you should be able to see that the Greatest Common Factor is 5.
Clearly, 10 is 5 times 2, 35 is 5 times 7, multiply those three factors, and we get 70, that is the Least Common Multiple, which is the Least Common Denominator. 70 is the Least Common Denominator, so we wanna get each of those fractions a denominator of 70. So that means that the first one will multiply by 7 over 7, the second one we'll multiply by 2 over 2, we'll get 7 over 70 minus 2 over 70, that gives us 5 over 70, which we can simplify to 1 14th.
And that is the actual answer that would appear on the test. Some further tips on the Least Common Multiple. If A is a factor of R, then the Least Common Multiple of a number, and its multiple of a factor, and the number of which it is a factor must be R. So for example, the Least Common Multiple of 8 and 24 has to be 24, that's very important.
Another important thing, if it's obvious that A and B have no factors in common greater than 1, in other words, the Greatest Common Factor is 1, then the Least Common Multiple would have to be their product, A times B. So for example, the Least Common Multiple of 7 and 15 must be 7 times 15. And again, this is the type of thing you should just be able to see, you shouldn't have to follow any procedure to see that 7 and 15 have no factors in common.
Now, we leave it to this, I have outlined a procedure here, DO NOT, DO NOT, DO NOT rely on this procedure as your only method of finding the Least Common Multiple, you MUST develop your intuition. This is a really important point. If your only strategy for the integer properties is memorizing procedures, you are absolutely guaranteeing that you will have a mediocre performance in Math on the test.
To have an above-average performance, you must develop some number sense. I'm going to recommend an exercise that will help build number sense related to these skills. Everyday, pick a pair of two-digit numbers. Perhaps you could generate them randomly, pick them out of a hand, or get a calculator to generate them, perhaps you could get a friend to pick them randomly for you.
Get the pair of numbers, then in your head, without writing anything down, do your best to figure out the Greatest Common Factor and the Least Common Multiple. When you have your best guesses, then follow the whole procedure outlined here to find these values to check yourself. If you do this with a friend or study partner, you can quiz each other and check each others work.
And of course the point is to develop your mental Math abilities, develop your intuition, so that sometimes you'll just see the Greatest Common Factor, you won't have to rely on a procedure to get it. In summary, we talked about what the Least Common Multiple is and how to find it from listing factors. We talked about how to find the Least Common Multiple using Greatest Common Factor.
We pointed out that the Least Common Multiple is the Least Common Denominator, and this simplifies fraction addition and subtraction. We talked about some further tips for the Least Common Multiple. And we outlined an exercise for building number sense in intuition for the Greatest Common Factor and the Least Common Multiple. And once again, one of the very best things you can do for yourself on the math section is to develop your number sense.
