Find the highest common factor, HCF, of 48 and 108.
So what we need to do is actually find the highest common factor. But what is the highest common factor? Well the highest common factor is actually the largest number that will go into both 48 and 108. So when I say go into, what I mean is that actually 48 and 108 could be divided by this number with no remainder. So the method I’m going to show you is actually one called prime factor decomposition. So we’re actually using our prime factors to find the highest common factor. You could also do this question by listing out the factors of 48 and 108.
So I’m gonna start with 48. And what we’re going to do is actually divide this by prime numbers. What I find helps is actually to write out the first few prime numbers just so we can see them when we’re doing this problem. And just a quick reminder to ourselves, a prime number is a number that has exactly two factors. So you can see that two could be divided by two and one, three by three and one, et cetera. So we know that there are prime numbers, so we just charted a few of these down to help us.
Okay, what we’re gonna do now is divide 48 by one of our primes. We usually start with one that we find fairly easy, so often it’s a good idea we can start with two. So we can say that 48 divided by two is 24. So we put a two, and we circle this because this is a prime factor. Then we’ve got 24. So then we can actually divide by two again. We could divide by three, but I’m just gonna carry on with the twos as we started with that. So 24 divided by two is 12, again circling our prime factor. So then we can actually divide by two once more, so 12 divided by two is six. So we’ve got two circled because that’s our prime factor. And then we’ve got six. Yes, again we could have divided here by three, but I am gonna do that when we get to the end. So then finally, we can divide six by two, and we’ve got two and three.
I’ve actually circled both of these because these are both prime factors. And we know that once we’ve actually got all of our factors circled, we know that we’ve reached the end because three and two cannot be divided by any other number. That’s a prime number. Okay, great so now we’ve actually divided 48 into its prime factors. And now we’re gonna do the same to 108. So again, I’ve actually started with two because it’s nice and straightforward for us to do. So 108 divided by two is 54. So again we’ve circled our two as it’s our prime factor. And then once more we divide it by two, so we’ve got 54 divided by two, which is 27. So we’ve got two circled as another prime factor, and then we’ve got 27.
This time we’ve got 27. So this can’t be divided by two, so we’re gonna move up and look at our next prime number, which is three. And it can be divided by three. So threes into 27 go nine times. So we’ve got three circled, and then we’ve got nine. And we’ve circled three again because it’s our prime factor. And then finally, three multiplied by three gives us nine. So that means we’ve got another two prime factors here. So again, we’ve reached the end because we know we can’t go any further because the last two factors we’ve got are both circled because they’re both prime.
Okay so we’ve now got our prime factors for 48 and 108. What do we do next? Well now what we do is we use a very useful mathematical diagram called a Venn diagram, and this is actually gonna help us to identify our highest common factor. So first of all, what we’re going to do is actually write in the center area the factors that are common to both 48 and 108. So first of all, we’ve got a two cause we’ve got a two in each. Then I’ve written in another two because there’s another two in each of our 48 and 108 prime factors. And then the final shared factor is a three because there’s a three in each of our prime factors. And then I’ve actually added the next two factors that are left over from our 48, so we’ve got two more twos. And then I’ve added the final two factors from 108, which are both threes.
So we’ve now got our completed Venn diagram, but how do we use this? Well the first way we could use this is actually to find the lowest common multiple. I know it’s not part of this question but it’s worth mentioning. And if we actually multiplied together all of the prime factors that we have, so two by two by two by two by three by three by three, this would give us the lowest common multiple of 48 and 108. But in this question, we’re not interested in that; we want the highest common factor. Well to find the highest common factor, what we actually do is multiply each of the numbers in the center section together because these are our shared factors. So therefore, the highest common factor would be equal to two multiplied by two multiplied by three, which should be equal to four multiplied by three because two multiplied by two is four, which will give us a highest common factor of 12. We’ve actually there solved the problem because the highest common factor of 48 and 108 is 12.
So this method is really good when we’re actually dealing with numbers that have a large number of factors, or also the question might ask you to use prime factors to find the highest common factor or lowest common multiple. However, as I said, we could use the alternative method of actually listing factors. So we’re gonna do that just to check our answer as well. So we’re gonna start with 48. So our first pair of factors is one, 48 cause one multiplied by 48 gives us 48. Then we’ve got two and 24, three and 16, four and 12. Then we know that five doesn’t go into 48 because 48 doesn’t end in a five or a zero. And then finally, we have six and eight because six multiplied by eight gives us 48. We don’t have seven because seven doesn’t go into 48 cause seven multiplied by seven is 49. And then we’ll be back round to eight which we’ve already got as one of our factors.
Okay, so now let’s find the factors of 108. So we have one and 108, two and 54, three and 36, four and 27. Again, five doesn’t go into 108 because 108 doesn’t end in a five or zero. So we’ve got six and 18. And then our final pair is nine and 12. And that’s because, again, seven and eight don’t go into 108. You have seven times 15 would give us 105 so it’s close, but it doesn’t go into it. And then 10 also wouldn’t go into 108 because 108 doesn’t end in a zero. And then finally, 11 also doesn’t go into 108. So then we are back round to 12.
So as you’ve seen, it’s actually quite difficult to find all of the factors and make sure we’ve got them all for 108. So that’s why the first method is very good for this type of question. So now what we need to do is actually inspect our lists. And we can see that actually the highest common factor, so the highest number that’s actually a factor in both for 48 and 108, is 12. So therefore, we’ve shown the alternative method and checked our answer.