Welcome to a lesson on determining the least common multiple of two terms.
The least common multiple or lowest common multiple.
Identified by LCM of two rational numbers a and b.
Is the smallest positive rational number: that is an integer multiple of both a and b.
Since the LCM is a multiple of a and b it can be divided by a and b without remainder.
And the LCM is the smallest number where this is possible.
Let's try to illustrate this idea lets say we want to determine the least common multiple of four and six.
what we could do is start listing multiples of 4 as we see here and then list multiples of 6.
Then what we could do is identified the common multiples or the numbers that occur in both of these lists.
So we see 12 occurs in both lists.
So does 24, so does 36.
48 and so on. So these would be the common multiples.
As we see list here, so the least common multiple of
4 and 6 ,is simply the smallest one of these.
Which would be twelve. As we see here and we also see up here.
So if we were determined the LCM or the least common multiple six and eight. We can start by listing multiples of 6.
And so on. Again notice we're just adding six each time or you could think of this as six,
times one times two times three,
and so on. Now do the same for eight.
8, 16, 24, 32,
40 and just keep listing these until we find a number that these two lists have in common.
And we want the smallest of those common multiples.
And we can see from our list 24 is the smallest number
that these two lists of multiples have in common. So the LCM
is equal to 24. The method riding out multiples only practical for determining the LCM of
small numbers. As a result we have a different way to build the LCM of
two numbers or terms. By analyzing a terms
prime factors and here's the key.
The least common multiple must contain all of the different factors.
In the terms and each factor must be repeated.
The maximum number of times it occurs in anyone of the terms.
So what we'll do is find all the prime factors that occur.
In each of the numbers and then use it the maximum number of times it occurs in any of the terms prime factorization.
Lets go ahead and give it a try.
We wanna determined the LCM 24 and 30 so we gonna write out the prime factorization of each of these.
24's eight times three I know 8 is two times two times two.
30 would be six times five and six is two times three.
Again if I lost you here the the way you can obtain the prime factorization,
would be to take one in the numbers.
And make a prime factorization tree 24's is eight times three.
Three is prime, 8 would be four times two.
2 is prime, four is times two.
So the prime factorization is two times two times two times three.
And we could do the same for 30 okay now here's the procedure to build the LCM.
We will first list all of the different factors.
From both prime factorizations notice the prime factorizations contain factors of 2, 3 and 5.
Now we're not done we have to determine.
The most number of times that any of these factors occur.
In either of the prime factorizations so what we mean by that is.
This has three factors of two and this has one factor of two.
We have to take the most number of times the two occurs.
And any of these prime factorizations that would be three times.
So we need 3 factors of two, next we look at the factor of 3.
It occurs once here and once here so we only need it once.
And now we'll look at the factors of 5 it occurs 0 times here and one time here so we need one factor of 5.
This is how we build the least common multiple 24 and 30 now we can multiply this out.
Should be two times two times two times three times five.
Which is equal to 120, so the LCM of 24 and 30 is equal to 120.
Lets go and try another, now notice on this example we're introducing variable factors but that's okay the procedures going to be exactly the same.
Lets go ahead and write these out in prime factored form.
So 27 x squared y squared would be three times three times three times two factors of x and two factors y.
If you're puzzled by where the three times three times three came from.
Again it came from making a factor tree for 27.
So we have 3 factors of 3. 15 would be three times five.
X cubed so three factors of x.
And one factor of y. So we'll now build the least common multiple.
We'll first list all the different factors that are used.
Let's get these factors first we have a factor of three and a factor of five.
We also have factors of x and factors of y.
So we have to include these factors now we're not done, we have to go back and determine.
The most number of times these factors occur.
In either of these factorizations, so we have three factors have three.
Here and one factor three here. We have to have three factors of three.
In our least common multiple.
Next there's 0 factors of five here and one here so we need one.
Now lets look at the factors of x, there's two factors of x here and three factors of x here.
We have to take the most that occur in either of the two factorizations.
So we need three factors of x, y occurs twice here and once here so we need it twice.
Be careful not to add the number of times the factor occurs.
You just take the most number it occurs and either one of the list so the result.
Would be 3 to the third, that would be 27 twenty-seven times 5 equals 135 and we have x cubed y squared.
Let's try another.
We'll begin writing out each term in prime factor form.
For sixty we have six times ten.
Six is two times 3, ten is two times five.
So we have two factors of two.
One factor of three and one factor of five we also have five factors of m and two factors n.
72 that would be eight times 9, 8 is four times two.
Two is prime, four is two times two and nine is three times three.
So we have three factors a two and two factors of 3.
As well as two factors of m.
And four factors of n.
Let's go ahead and build our LCM.
Let's first lists all the different factors that are used.
That would be a factor of two, 3,
5, m and n. Now let's go back and determine how many of these factors we need.
This has two factors of two, this has 3
so we need 3. This has one factor of three and this has two.
So we need two. This has one factor of 5 this has 0 factors of 5 we need one factor of 5.
We have five factors of m here.
And two here so we need five. We have two factors of n here and four here.
So we need four factors of n. So two to the third
that would be 8. Three square would be nine, eight times nine is,
72 times 5 that's 360.
M to the fifth and
n to the fourth would be our LCM.
I hope you found this video helpful. Thank you for watching!