Professor Dave here, let’s learn about least common multiples.
As we begin to learn more about numbers, there are certain ways in which we will want to
compare two numbers.
One concept we must learn is that of a multiple.
Three times one is three, three times two is six, three times three is nine, three times
four is twelve, so these are examples of multiples of three.
They are the numbers you get when you multiply three by all the positive integers, and the
list goes on, all the way to infinity.
What about for the number four?
Well that would give us four, eight, twelve, sixteen, and so on, and we could construct
multiplication tables like these as we go through all the numbers.
So what if we compare three and four, we might want to know the smallest number that is in
both of these lists.
We would call this the least common multiple, because it the smallest multiple that is common
to both numbers.
In this case, that’s twelve.
Let’s list the multiples of five now.
Five, ten, fifteen, twenty, twenty-five.
What is the least common multiple of three and five?
That would be fifteen.
What about four and five?
That would be twenty.
As we can see, many times the least common multiple is simply the product of the two
numbers.
This product will always be a common multiple, because the product of the two numbers must
necessarily have the two original numbers as factors.
However, this will not always be the least common multiple, sometimes there is a smaller
common multiple.
Take six and nine.
Six times nine is fifty-four, but if we write out the multiples for both numbers, we find
that already at eighteen we get the least common multiple.
So when looking at smaller numbers, it is a good idea to write out the multiples to
find the answer.
Once we get to some bigger numbers, this method may get tedious, so there is a bit of a shortcut.
Take the numbers twelve and eighty.
The first step will be to find the prime factorization of each number.
For twelve that’s two times two times three, and for eighty that’s two times two times
two times two times five.
Now, we simply take every factor we see here and multiply them together, with each factor
being multiplied a number of times equal to the most times it appears for one number.
Two appears four times for eighty, which is more than the two times it appears for twelve,
so let’s put four twos, and then the one three, and then the one five.
We find the product, and that’s two hundred forty, so this will be the least common multiple
for twelve and eighty.
What is one application of this concept?
Well, we learned about fractions, and sometimes we might want to know which of two fractions
is larger, which can be difficult if they have different numbers in the denominator.
Take one third and one fourth, for example.
It’s true that if we represent these fractions as sections of a circle it is pretty easy
to see that one third is larger, but this becomes even more obvious if we alter these
fractions so that they do have the same denominator, by finding the least common multiple.
For three and four, that’s twelve, so what do we need to do to get twelve on the denominator
for both of these fractions?
For one third, we can multiply both top and bottom by four.
If we do this, we are not altering the magnitude of the fraction, because four over four is
one, and anything times one is itself, but we can change the way the fraction is expressed.
Now we have four twelfths.
For one fourth, we can multiply by three over three, giving us three twelfths.
Now that they have the same denominator, it is easy to see that four is bigger than three,
so four twelfths must be bigger than three twelfths.
Although it was already obvious in this instance, we can use this technique for much trickier
fractions so that we can be sure as to which one is larger.
Let’s check comprehension.