Danny is buying some fireworks for a local event. Catherine wheels are sold in packs of five. Whizzbangs are sold in packs of 12. Firestars are sold in packs of eight. Danny wants to buy the same number of Catherine wheels, whizzbangs, and firestars. What is the smallest number of boxes of each type of firework that Danny can buy?
So if we’re looking here at what we’re trying to find out, well, we’re trying to find out the smallest number of boxes of each type firework that Danny can buy. And because each firework comes in different pack size, what we’re looking for is the lowest common multiple of eight, 12, and five. And first of all, we need to know what the lowest common multiple is. Well, the lowest common multiple is the lowest number that appears in each of the eight, 12, and five times tables. Now, what we could do is list the multiples of eight, 12, and five. That’s the first method we could use. The other method we could use is using prime factor decomposition.
Well, what I’m gonna do is I’m gonna show you the method that uses prime factor decomposition because this is more useful because it can be used for all different types of numbers, even though they’re much bigger than the numbers we’ve got here. And then we can have a look at the other method to check out the answer. Well, first of all, if we take a look at five, five is a prime number and a prime number is the number that can only be divisible by itself and one and it has exactly two factors, and five has that. So therefore, we don’t have to use prime factor decomposition any further with five.
So now if we’re gonna work out the prime factors of eight, what we need to do is find a prime number that goes into eight. But what I usually do is list the first few prime numbers to help me. So we’ve got two, three, five, seven, 11, 13, and 17. So we have two is our first prime number. And this is very special because it’s the only even prime number. It’s also worth noting that one is not a prime number. And it’s common mistake to put it down as one because one even though it has a factor of itself of one, it does not have two factors; it only has the one factor itself because itself is one. So the first thing we do when we’re using prime factor decomposition is divide eight by a prime number. Well, the prime we know that goes into eight is two. So we divide eight by two. So we have two and four. And what we do is we circle the prime number. So I’ve done that right here.
And then next, what we do is divide by prime number once more. When we do that, we get four divided by two, which is two. So now as you can see, each of our branches is a prime number. And we’ve circled each of them. So at this point, we would stop. And then we do the same for 12. So if we divide by a prime number, two, we’ll get two and six. And we circle thee two again cause this is a prime number. We could have divided by three because three is a prime number and that would have led us to the same answer. But I have chosen to do it by two. And then again, with six, we could choose to divide six by two or three because six divided by two is three or six divided by three is two. So we’ve now reached the final branches because, again, all three branches here are prime factors. So I’ve circled them all.
So now, what we’re gonna do is use a Venn diagram to help us work out the lowest common multiple. And each section in the Venn diagram is going to help us represent either the factors of five, 12, or eight. And we’re gonna use the prime factors here. So I’m gonna start with factors of five. Well, the only prime number that’s a factor of five is five. So we put this into our Venn diagram. And it goes into this section here because it’s not in 12 as well or eight as well. So therefore, it stays in the section that is only five.
So now, we look for any common prime factors between eight and 12. And you could see that both eight and 12 have two as one of their prime factors. So we put two in the overlapping section between eight and 12. It does not go into the middle section because two is not a factor of five. And then we could see that we have another shared factor because two is in both of the factors of 12 and eight once again. So now, we have two twos in that section. So now, we have one more two as a prime factor of eight. So this goes in the section that is only eight. And then we have three, which is a factor of only 12.
Okay, great, so we’ve now put all of our prime factors into our Venn diagram. But how does this help? Well, we can use Venn diagrams to help us find the highest common factor although it’s common multiple. The highest common factor would in fact be the central part of our Venn diagram. However, that would not be a highest common factor of five, 12, and eight because there’s nothing in the central compartment. But we’re not looking for the high common factor; we’re looking for the lowest common multiple. So how do we find that?
Well, we find the lowest common multiple by multiplying all of the factors that we have within our Venn diagram together. So therefore, the lowest common multiple is gonna be equal to five multiplied by two multiplied by two multiplied by two multiplied by three. So we’ve now multiplied each of the terms within our prime factors in our Venn diagram. It is worth noting though that this is different than just multiplying all the prime factors of five, eight, and 12. And that’s because if we count the prime factors, we would have seven. However, we’re only multiplying together five prime factors. And this is because two of them are shared as we can see from our Venn diagram.
So now, if we multiplied them together, we’re gonna get five multiplied by two, which is 10, multiplied by another two, which is 20, multiplied by another two, which is 40. And then we’re gonna have multiplied by three. So we get 40 multiplied by three. Well, if we multiply 40 by three or four by three, it’s 12. So we’re gonna get 120. So we found the lowest common multiple is 120. We thought at the beginning that we could check this out by writing out all the multiples of five, 12, and eight until we’ve found a number that was the same. But as you can see, because the lowest common multiple is 120, we’d have to write out a lot of multiples. To get to this stage, that’ll be time-consuming and it would not be the best method.
So now, what we need to do is work out how many packets of Catherine wheels, whizzbangs, and firestars. Danny needs to buy. So first of all, we’ll start with Catherine wheels. These are sold in packs of five. So therefore, to work out how many packs we need, we need to divide 120 by five. So I’m gonna do that using short division. So five doesn’t go into one. So we write zero and carry the one. So then we see how many fives go into 12 which is two. And then we carry the two. Then we see how many fives are going to 20 which is four. So we get 24. So therefore, we can say because 24 multiplied by five is 120, we’re gonna want 24 boxes of Catherine wheels.
Well next, we’re gonna have a look at whizzbangs. And these are sold in packs of 12s. So we need to see how many 12s are going to 120. Well, we could say it’s just going to be 10 because 10 12s are 120. But we can double-check this using our method that we used just now, which was short division. Well, 12s into one don’t go, so we carry the one. So we do 12s into 12, which is one, and then 12 into zero is zero. So we get 10. So as 10 multiplied by 12 is 120, therefore, we’re gonna have 10 boxes of whizzbangs.
Then finally, we have firestars. And to work this out, we do 120 divided by eight. Well, eight into one don’t go, carry the one. Eight into 12 go once remainder four, and then eights into 40 goes five times. So we’ve got 15 multiplied by eight equals 120. So therefore, we’ve got 15 boxes of firestars. So therefore, we can say that the smallest number of boxes of each type of firework that Denny can buy are 24 boxes of Catherine wheels, 10 boxes of whizzbangs, and 15 boxes of firestars. And it’s is gonna have 120 of each firework.