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.
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.
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.