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# What Is the Least Common Multiple?

What is the least common multiple of two or more numbers? What does it have to do with finding the lowest common denominator of a pair of fractions? Keep on reading to find out!

By
Jason Marshall, PhD,
November 30, 2012
Episode #131

How do you find the lowest common denominator of the fractions 1/5 and 3/4? Well, all you have to do is find the least common multiple of the denominators of those fractions. Okay, but what exactly is a least common multiple? And, once you've found it, how exactly does it lead to the lowest common denominator? For that matter, since we've already learned about common denominators, why do we also need to learn about lowest common denominators? Whether you need a full-blown refresher on all things LCM and LCD or just a little tune-up, keep on reading because those topics are precisely what we’ll be talking about today and over the next few weeks.

## Recap: What Are Common Denominators?

Let's start our journey into the realm of least common multiples and lowest common denominators with a quick recap of good old basic common denominators. As we've learned, two fractions that share the same denominator—such as 1/4 and 3/4 or 7/16 and 3/16—are said to be written in terms of a common denominator. Why all the fuss over this seemingly simple idea? Well, as we'll talk about in much more detail over the next few weeks, once you've written fractions in terms of a common denominator, it's a lot easier to work with them.

So how do you find a common denominator? As we learned last time, the quick and dirty way to find a common denominator of two fractions is to multiply the numerator and denominator of each by the denominator of the other. So we can rewrite 1/6 and 2/3 in terms of the same common denominator by multiplying the top and bottom of 1/6 by 3 (the denominator of 2/3) to get 3/18, and the top and bottom of 2/3 by 6 (the denominator of 1/6) to get 12/18. Once you've got the hang of it, rewriting fractions in terms of a common denominator like this is relatively efficient and effective. So then why isn't this the end of our story?

## Common Denominators are Very Common

In truth, there's absolutely nothing wrong with using this method to find common denominators. But it's important to realize that every pair of fractions has more than one possible common denominator. In other words, the fractions 1/6 and 2/3 (or any other pair) don't have one-and-only-one all-powerful common denominator. In fact, they have many, many possible common denominators. And, as it turns out, some of them are easier to work with than others.

For example, since we know that multiplying the top and bottom of a fraction by some number doesn't change the ratio represented by the fraction, we could have found a different common denominator for the fractions 1/6 and 2/3 by multiplying their tops and bottoms by 10 times the denominator of the opposite fraction instead of its actual value. Which means we could have multiplied the top and bottom of 1/6 by 30 (which is 10 x 3) and the top and bottom of 2/3 by 60 (which is 10 x 6) to get equivalent fractions of 30/180 and 120/180. As you can check by simplifying these fractions, they are indeed equivalent to the ones we started with. And they're certainly both written in terms of a common denominator. But that common denominator, 180, is kind of a big number…and it makes working with these fractions somewhat less than pleasant. Which is exactly where the idea of the lowest common denominator—and its companion the least common multiple—enters our story.

## What Is the Least Common Multiple?

The lowest common denominator—often abbreviated LCD—is exactly what it sounds like. Out of the infinite number of possible common denominators (yes, if you think about it, you'll realize that there are indeed an infinite number of them), the lowest common denominator is the particular common denominator that is smallest. As I mentioned earlier, to find the LCD of a pair of fractions, you first need to find what's called the least common multiple—aka, LCM—of their denominators. What’s the least common multiple? Well, a multiple of a number—let's say 3—is any number that can be made by multiplying that number—in our case 3—by any other whole number. For example, since 3 x 2 = 6, we say that 6 is a multiple of 3. Since there are an infinite number of whole numbers, every number has an infinite number of multiples. In addition to 3 and 6, the numbers 9, 12, 15, 18, and so on are also multiples of 3.

Once you know how to find the multiples of a number, finding the least common multiple of two or more numbers is easy: Simply figure out what all of the multiples of each of the numbers are and then find the lowest multiple that they all have in common. For example, to find the least common multiple of the numbers 3 and 4, start by figuring out what all the multiples of the numbers 3 and 4 are. As we found earlier, the multiples of 3 are all the numbers like 3, 6, 9, 12, and so on. And, as you can work out, the multiples of 4 are all the numbers like 4, 8, 12, 16, and so on. So, what's the least common multiple of 3 and 4? It's the smallest multiple that these two numbers have in common. Which, in this case, is the number 12.