Let’s look at least common
multiple, what it is, and how we would find it. Least common multiple, well first
it’s a multiple. A multiple is any number multiplied
by an integer. Or put another way, a number that
may be divided by another number with no remainder.
Here are some examples of some
multiples. Multiples of three are three, six,
nine, twelve, fifteen. Multiples of twelve would be
twelve, twenty-four, thirty-six, forty-eight.
But we’re not just looking for the
multiple, we’re looking for something that is common. It’s shared by two or more
numbers. It’s also going to be the least,
and by that we mean the lowest number.
So we would say the least common
multiple is the smallest number that is a multiple of two or more numbers.
Here’s an example: find the least
common multiple of four and six. When we’re working with least
common multiple, we usually write LCM, open parentheses, then we enter the numbers
that we wanna find the least common multiple of. That’s the format that you’ll
usually find the least common multiple in.
To solve this problem, we wanna
find the prime factorization of both four and six. four is made up of two times two,
and six is made up of two times three.
For something to be a multiple of
four, it needs two and a two. So whatever the least common
multiple is, it’s going to need two times two in it.
For something to be a multiple of
six it needs a two. And we have already included one
two, so we don’t have to add another one. But we haven’t added a three yet,
so we’ll need to do that. two times two times three will be our least common
multiple of four and six. We know that this number will be a
multiple of four because it has two times two in the factorization, and we know that
it’ll be a multiple of six because it has three times two in the factorization.
So our least common multiple here
is twelve. That means that twelve is the
smallest common multiple of four and six.
So maybe you’re thinking, “Well I
could have figured that out with all of that factoring.” That’s true. There is another way to do it.
You could list all the multiples of
both of those and then try to find the smallest one. And here, that’s twelve. But it’s not always the best
way. The reason that’s not always best
is you could end up with a problem like this, the least common multiple of one
hundred and thirty-seven, fifty-three, and five hundred and seventeen. You certainly wouldn’t wanna waste
your time listing out each of the multiples. Instead, it’s always best to look
for the prime factors and then find the multiple.
Speaking of bigger numbers, here’s
a few that are slightly bigger.
What’s the least common multiple of
thirty-nine and thirty-two? We wanna solve this problem by
factoring so let’s start there. I recognize that thirty-nine is
divisible by three; thirty-nine is three times thirteen. Both of those are prime numbers so
we’ll stop there. Thirty-two is divisible by two and
sixteen, sixteen is two times eight, eight is two times four, and four is two times
When you look at the factorization
of thirty-nine and thirty-two, you’re going to realize that they actually share no
common factors, which means to find a multiple that they have in common, we need to
multiply three times thirteen times two times two times two times two times two. Or you can also just multiply
thirty-nine times thirty-two. The least common multiple for
thirty-nine and thirty-two is one thousand two hundred and forty-eight.
We’re going to use the same process
to find the least common multiple of three numbers that we did for two. We need to find the factorization
of all three of these numbers. Let’s start with the four. That’s two times two. And then the six, two times
three. And finally with forty-nine, we
find it’s seven times seven.
We know we’ll need two times
two. Since we’ve already added a two, we
now only need to add a three for our six. But the factors of forty-nine are
two sevens, and we’ll need to include both of those.
The least common multiple of four,
six, and forty-nine will be two times two times three times seven times seven for a
final answer of five hundred and eighty-eight.
Here’s our last example, the FIFA
World Cup takes place every four years and the US census is taken every ten years
both of them took place in two thousand and ten. When is the next year in which both
of these events will happen again? Let’s pull out the key information
here, which is the World Cup takes place every four years and the US census is taken
every ten years. So we wanna know what are
multiples, what is in fact is the least common multiple of four and ten?
We’ll follow the same procedure,
finding the factors of four and ten. We know four is two times two and
ten is two times five.
We’ll need two times two and also
that five factor. Two times two times five is
twenty. The least common multiple of four
and ten is twenty, but that’s not the answer to the question.
The question is when is the next
year in which both of these events will happen. So we need to know what should we
do with that twenty. That twenty means that every twenty
years, the four and the ten will line up. So we need to add twenty years to
two thousand and ten, which is two thousand and thirty. two thousand and thirty is
the next year that the World Cup and the US census will happen at the same time.
So now you know, and you’re ready
to use these skills to find the least common multiple of lots of numbers.