Using Prime Factorizations to Find the Least Common Multiples

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  • 0:01 Least Common Multiple
  • 2:03 Prime Factorization
  • 3:24 Finding the LCM
  • 4:52 Practice
  • 6:58 Lesson Summary
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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.

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