Login

Using Prime Factorizations to Find the Least Common Multiples

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: Equivalent Expressions and Fraction Notation

You're on a roll. Keep up the good work!

Take Quiz Watch Next Lesson
 Replay
Your next lesson will play in 10 seconds
  • 0:01 Least Common Multiple
  • 2:03 Prime Factorization
  • 3:24 Finding the LCM
  • 4:52 Practice
  • 6:58 Lesson Summary
Add to Add to Add to

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Login or Sign up

Timeline
Autoplay
Autoplay
Create an account to start this course today
Try it free for 5 days!
Create An Account

Recommended Lessons and Courses for You

Lesson Transcript
Instructor: Jeff Calareso

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

Finding the least common multiple can seem like a lot of work. But we can use prime factorization as a shortcut. Find out how and practice finding least common multiples in this lesson.

Least Common Multiple

Let's talk about the least common multiple. Think about babies. Most people have one baby at a time. Some people have multiples, like twins. As multiples go, twins are pretty common. It's like a 2-for-1 sale on kids! Plus, each kid has a built-in best friend and/or partner in crime. Then there are triplets. When people have triplets, they need bigger cars. What about quadruplets? That's one of your less common multiples. Same with quintuplets.

When we talk about least common multiples, we're not really asking how many quintuplets you know, though, probably not many, right? The least common multiple of two or more numbers refers to the smallest whole number that is divisible by those numbers. So we're not looking for the least common multiple as in the most rarely occurring. Rather, we want the least common multiple, as in the smallest shared multiple.

Imagine the birthday party for those quintuplets. Maybe we need party favor bags; the shrieking whistles come in packs of 15, the permanent markers come in packs of 32, and the matchbooks come in packs of 45. First of all, those are terrible party favors for a kid's birthday party. But more importantly, how many of each will you need to buy so that you have an even number? This is where knowing the least common multiple is useful. Before we tackle that problem, let's start simple.

Let's say we have 3 and 5. To find the least common multiple, we just start listing the multiples of each. The multiples of 3 are 6, 9, 12, 15, 18, 21, 24, 27, 30, and so on. The multiples of 5 are 10, 15, 20, 25, 30, and so on. What multiples are shared? Well, of the ones we listed, there's 15 and 30. What's the smallest? 15. So 15 is the least common multiple of 3 and 5. By the way, if you had 15 kids, you'd need a bus.

Prime Factorization

With small numbers like 3 and 5, listing the multiples is perfectly fine. It's kind of like how with twins going out to a restaurant isn't too big of a deal. But what about with more challenging numbers? What if we have 36 and 40? This is more like taking quintuplets to a restaurant. No one will be very excited sitting next to them. This is where prime factorization comes in.

What is prime factorization? This is when we're finding the prime numbers that multiply together to make a number. Let's unpack that a bit. The factors of a number are the numbers that, if you multiply together, you get the original number. Some factors of 36 are 3 and 12. Why? Well, 3 x 12 is 36. Other factors are 2 and 18. 6 x 6 also gives us 36. Prime factors are factors that are prime numbers, or numbers larger than one that can only be evenly divided by one or themselves.

We can draw a factor tree to find the prime factors. You can start a tree with any factors. Let's start with 3 and 12. Well, 3 is a prime number, so that branch stops pretty quickly. What about 12? Well, 3 and 4 are factors of 12 and 2 and 2 are factors of 4, so the prime factors of 36 are 2 x 2 x 3 x 3.

Finding the LCM

We can use this information to find the least common multiple by following three steps. First, complete the prime factorization for each number. We just did that for 36. Let's do it for 40. 40 is 2 x 20, 2 is prime. So, 20 is 2 x 10 and 10 is 2 x 5, so 40's prime factors are 2 x 2 x 2 x 5. Now we're done with step one.

Second, find which prime number occurs most often. Most often? That seem weird? A little, yeah, but stick with me. List these numbers out. So both sets have 2s, but there are more 2s with 40; that's what I mean by 'occurring most often.' Then there are two 3s with 36 and one 5 with 40. In these cases, that's most often because they don't occur in the other sets of factors.

Third, find the product of these numbers. Okay, we're almost there. This will be cool. So, 2 x 2 x 2 x 3 x 3 x 5. What is that? Well 2 x 2 is 4, 4 x 2 is 8, 8 x 3 is 24, 24 x 3 is 72, and 72 x 5 is 360. Guess what? 360 is the least common multiple of 36 and 40. We can make sure it's a multiple by dividing each number into it. 360 / 36 is 10. 360 / 40 is 9.

To unlock this lesson you must be a Study.com Member.
Create your account

Register for a free trial

Are you a student or a teacher?
I am a teacher

Unlock Your Education

See for yourself why 30 million people use Study.com

Become a Study.com member and start learning now.
Become a Member  Back

Earning College Credit

Did you know… We have over 95 college courses that prepare you to earn credit by exam that is accepted by over 2,000 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it free for 5 days!
Create An Account
Support