Back To Course

Algebra I: High School20 chapters | 168 lessons | 1 flashcard set

Watch short & fun videos
**
Start Your Free Trial Today
**

Start Your Free Trial To Continue Watching

As a member, you'll also get unlimited access to over 70,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Free 5-day trial
Your next lesson will play in
10 seconds

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.

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.

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.

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.

Let's practice with another set of numbers. What if we have 14 and 25? What's step one? Prime factorization. Do the tree: 14 is 2 x 7 and, well, those are prime. 25 is 5 x 5 - more prime numbers. Neat. Step two is find the numbers that occur most often. That's easy here - there are no repeats. Onto step three: finding the product of these numbers. 2 x 7 x 5 x 5. That's 350. So 350 is the least common multiple of 14 and 25.

What if we add three numbers? It's time for our party favor example. We add 15 shrieking whistles, 32 permanent markers, and 45 matches. What's the least common multiple? Let's start with prime factorization. 15 is 3 x 5; easy. 32 is 2 x 16, 16 is 2 x 8, 8 is 2 x 4, and 4 is 2 x 2, so 32 is 2 x 2 x 2 x 2 x 2. Okay, 45: that's 5 x 9 and 9 is 3 x 3. Step two, find the most popular prime factors. Both 15 and 45 have 3s, but 45 has more, so we'll take those. They both have 5s too, so let's just take one of those. Now let's take our 2s from 32. So we have 2 x 2 x 2 x 2 x 2 x 3 x 3 x 5. Now for step three. We know those 2s equal 32, so 32 x 3 is 96, 96 x 3 is 288, and 288 x 5 is 1,440. So the least common multiple of 15, 32, and 45 is 1,440. That was way simpler than listing out every multiple for all three numbers until we found 1,440, right? It's also a good sign that we should find different party favors.

To summarize: we learned how to find the least common multiple for a set of numbers. This is the smallest whole number that is divisible by all the numbers. First, we complete the prime factorization of each number. Remember the factor tree: we're trying to break a number up into its prime number factors. Second, we find which prime number occurs most often and list these out. Finally, we find the product of all these numbers. That's going to give us our least common multiple.

After watching this lesson, you should be able to recall and demonstrate the steps required to find the least common multiple.

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

Create your account

Already a member? Log In

BackDid 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

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.

You are viewing lesson
Lesson
3 in chapter 13 of the course:

Back To Course

Algebra I: High School20 chapters | 168 lessons | 1 flashcard set

- What is Factoring in Algebra? - Definition & Example 5:32
- How to Find the Prime Factorization of a Number 5:36
- Using Prime Factorizations to Find the Least Common Multiples 7:28
- Using Fraction Notation: Addition, Subtraction, Multiplication & Division 6:12
- Factoring Out Variables: Instructions & Examples 6:46
- Combining Numbers and Variables When Factoring 6:35
- Transforming Factoring Into A Division Problem 5:11
- Factoring By Grouping: Steps, Verification & Examples 7:46
- Go to High School Algebra: Factoring

- NCLEX Information Guide
- TEAS Information Guide
- HESI Information Guide
- Business 329: Retail Operations
- Computer Science 320: Digital Forensics
- Messaging in Business Communication
- Retail Market Selection
- Retail Merchandise Management
- The Study of Retail
- Retail Sales Operations
- How to Study for the AFOQT
- GACE Test Day Preparation
- GACE Test Retake Policy
- GACE Test Accommodations
- NES Tests in New Mexico
- NES Test Day Preparation
- NES Test Cost

- British Associationism: History, Theories & Examples
- Nurse Aide: Definition & Duties
- Common Final Velocity in Inelastic Collisions
- How to Write Sets in Roster Form
- Converting Positive Integer Values in the Binary Numerical System
- Practical Application for Scientific Measurement & Dimensional Analysis
- Employee Management & Customer Experience: Relationship & Examples
- Operating Income within a Healthcare Organization
- Quiz & Worksheet - Macbeth Act 4, Scene 2
- Quiz & Worksheet - Calculating Osmolality
- Quiz & Worksheet - Creating Online Presentations
- Quiz & Worksheet - Researching User Experiences
- Quiz & Worksheet - Overview of La Noche Boca Arriba
- International Law & Global Issues Flashcards
- Foreign Policy, Defense Policy & Government Flashcards

- Essay Writing: Help & Tutorial
- Pathophysiology Textbook
- AP Macroeconomics Textbook
- Common Core ELA Grade 7 - Speaking & Listening: Standards
- Guide to Becoming a Substance Abuse Counselor
- ORELA Math: Geometric Solids
- Holt World History - Human Legacy Chapter 24: Nationalism in Europe
- Quiz & Worksheet - Features of Good Writing
- Quiz & Worksheet - Life & Writings of T.S. Eliot
- Quiz & Worksheet - Prejudice Theories and Ideas on Origins
- Quiz & Worksheet - Sexual Development and Maturation in Adolescence
- Quiz & Worksheet - Audience Opposition in Essays

- Unit Circle: Memorizing the First Quadrant
- Cell-Free Protein Synthesis: Steps & Applications
- Finding GRE Test Centers and Dates
- Cool Science Facts
- Best Apps for the Classroom
- ACT English Tips
- Fun Math Games for 1st Grade
- How To Check AP Exam Scores
- How to Pass the PSI Real Estate Exam
- How to Pass the RICA
- FTCE Social Science 6-12: Passing Score
- How to Pass Math

Browse by subject