# How To Solve Lcm Problems

During a car race, two competitors will stop to fill up the fuel tank.

One does it every 4 laps, and the other every 5 laps.

So this right over here is the least common multiple of 18 and 12. Here, you'll be able to do it a little bit more systematically, and you'll know what you're doing.

So, notice, now this number right over here has a 2 times 2 times 3 in it, or it has a 12 in it, and it has a 2 times 3 times 3, or an 18 in it. Really, really, really large and hairy numbers where you keep trying to find all the multiples, you might have to go pretty far to actually figure out what their least common multiple is.

Luis's teacher also assigns three projects per year. And we could keep going on and on looking at all the multiples of 30. Will William's teacher, after the first test, they're going to get to 24 questions.

## How To Solve Lcm Problems

Even though the two classes have to take a different number of exams, their teachers have told them that both classes-- let me underline-- both classes will get the same total number of exam questions each year. So this is probably a hint of what we're thinking about. We want the minimum multiples or the least multiple. Then they're going to get to 48 after the second test. They're going to get to 96 after the fourth test. So another way to come up with the least common multiple, if we didn't even do this exercise up here, says, look, the number has to be divisible by both 30 and 24. And say, well in order to be divisible by 24, its prime factorization is going to need 3 twos and a 3. And we already have 1 two, so we just need 2 more twos. So this makes it-- let me scroll up a little bit-- this right over here makes it divisible by 24. She wants to use all of the binders and pencils to create identical sets of office supplies for her classmates.The situation is shown below4, 8, 12, 16, 20, 24, 28, 32, 36 40, 44, 48, 52, 56, 60The other competitor will refill the tank after 5 laps, 10 laps, 15 laps, etc...The situation is shown below5, 10, 15, 20, 25, 30, 35, 4045, 50, 55, 60The number in bold are what they have in common.They will meet at the gas station after 20 laps, 40 laps, 60 laps, etc..The least number they have in common is 20We can also call this number the least common multiple.And you see the point at which they have the same number is at 120. They both could have exactly 120 questions even though Luis's teacher is giving 30 at a time and even though William's teacher is giving 24 at a time. If you take a two away, you're not going to be divisible by 24 anymore. If you take a three or a five away, you're not going to be divisible by 30 anymore. And it's also dealing with dividing these things. So what's the largest number of prime numbers that are common to both factorizations? Then you don't have a three times anything else. So this is essentially telling us, look, we can divide both of these numbers into 3 and that will give us the largest number of identical sets. So we've answered the question is 3, but just to visualize it for this question, let's actually draw 21 binders. So if there are three people that are coming into this classroom, I could give them each seven binders and 10 pencils. And we essentially are just thinking about what's the number that we can divide both of these sets into evenly, the largest number that we can divide both of these sets into evenly.And so if you were to multiply all these out, this is 2 times 2 times 2 is 8 times 3 is 24 times 5 is 120. We want to divide these both into the greatest number of identical sets. 30 is, let's see, it's 3-- actually, I could write it this way-- it is 2 times 15. So let's say the 21 binders so 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21. But that's the greatest number of identical sets Umama can make. So there's a couple of ways we could think about it. So what's the largest number that divides into both of them? We could list all of their normal factors and see what is the greatest common one. So let's just do the prime factorization method. And then 30 pencils, so I'll just do those in green. Let's think about what the greatest common divisor of both these numbers are. When the race begins, after how many laps will they meet at the fuel station, if they travel at the same speed?Solution The first competitor refill the tank every 4 laps The first competitor will refill the tank after 4 laps, 8 laps, 12 laps, etc...

## Comments How To Solve Lcm Problems

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