>> We're going to determine the least common multiple or LCM of each pair of numbers
which means you want to find the smallest number that is a multiple of both 16 and 18
and the smallest number that is a multiple of 72 and 54.
There are two main ways of determining the LCM.
One way is to listen multples of each number
and the from the list determine the smallest number that is common in both list.
And the other method is to build the LCM from the prime factorization.
Using the prime factorizations, if we multiply the unique prime factors of each number raised
to the power that is the most number of times that prime factor occurs
in either list we'll give us the LCM.
In this video we'll use the prime factorization method to determine the LCM.
So we'll start by determining the prime factorization of 16 and 18.
So the prime factorization of 16 consists of for factors of two and then for 18,
the prime factor of this will be 2 times 3 times 3.
Let's good ahead and list these.
Now to build the LCM of 16 and 18 it must contain factors of 2 and factors of 3.
These are the unique prime factors from both lists.
But now we have to raise these factors to the power that would be the most number
of times the factor occurs is in either list.
So notice that two occurs four times in the prime factorization 16 and one time
in the prime factorization of 18.
So now the most 18 times it occurs in your list is 4 times so to the 4th power has
to in the least common multiple of these two numbers.
Let's go ahead and highlight those.
Then notice that three occurs zero time in 16 but twice in 18 so we must have to factors
of an the LCM were 3 to the second power.
This product will be the LCM of 16 and 18.
To the fourth is 16.
3 to the second is nine so the LCM of 16 and 18 and 144.
Now let's go ahead and try the same for 72 and 54.
Now if I lost you on this prime factorization tree you may want
to review the video that goes over how to do this.
So for 72, we'll start with 8 times 9 and 4 times 2
and 2 times 2 and nine is three times three.
So to keep track of all these we have one, two, three, factors of 2 and two factors of 3 for 72.
54, let's start with 9 times 6, 3 times 3 and 2 times 3.
So we have one factor of 2 and three factors of 3.
Let's go ahead and list them again.
72 we have one, two, three twos and two threes and for 54 we have one, two and three threes.
So to build the OCM I'm going to have to have factors of 2 and again factors of 3.
And I will determine the most number times these factors occur in either list.
Notice there are three factors of 2 here and only one factor of two here.
So we'll need three factors of 2 in the LCM so raised to the third power
and now let's look at the factors of three.
There's two here and three here so we take the most number of times it occurs.
Occurs 3 times here so the LCM must contain the factor of three to third.
So here we have 2 to the third which is 8 times three to the third
which is 27 and 8 times 27 is equal to 216.
So 216 is the smallest number that's a multiple of both 72 and 54 which is called the LCM. ------------------------------a18c1bea5e3a--