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Prime factorisation: highest common factor and lowest common multiple

In this video Paula and Michael show an alternate method of finding the LCM and the HCF using prime factor trees and a Venn diagram
PAULA KELLY: OK, so you’ve seen how we can write a range of different numbers as a product of prime factors. So think about why we do this. How can we use our numbers written as products of primes?
MICHAEL ANDERSON: So prime factorization is really useful when we’re trying to find the highest common factor and the lowest common multiple of two numbers. So let’s have a look at two numbers, say 18 and 30.
PAULA KELLY: So let’s look at them initially, it’s not immediately obvious what the lowest common multiple would be, because you write out our 18 times table, our 30 times table–
MICHAEL ANDERSON: And see which one appears first. And similarly, we could write all the factors of 18 and 30 down and see which the highest number is that they have in common. But let’s look at it in another way. So we’re going to do a prime factor tree for 18. So how do you want to split 18 up?
PAULA KELLY: With even. So shall we start with a 2?
MICHAEL ANDERSON: Yes, 2, and that’s going to be a 9, and 2 is prime.
PAULA KELLY: 9, it’s not even, so you can’t have 2, so our next prime number 3.
MICHAEL ANDERSON: Yeah, and 3, 3’s a 9. So we can write 18 as 2 multiplied by 3 multiplied by 3. So let’s do the same for 30. How do you want to start splitting that one up?
PAULA KELLY: Well it is a multiple of 5, but we often start with our lowest primes. So should we start with 2 again?
MICHAEL ANDERSON: OK, so 2 and 15.
PAULA KELLY: 15 we know isn’t even, can’t be 2. Let’s have 3 and 5.
MICHAEL ANDERSON: OK, so 3 multiplied by 5 gives us 15. So 30 is equal to 2, multiplied by 3, multiplied by 5.
PAULA KELLY: OK so now we have a products of primes. How could we find our lowest common multiple or highest common factor?
MICHAEL ANDERSON: So let’s have a look at the highest common factor first. They’ve got a few prime factors in common, but actually the thing that we’re going to do next is draw a Venn diagram to display these prime factors in a diagram. So I’m going to draw a circle, and in the circle I’m going to put all of the prime factors of 18. And in this circle, I’m going to put all of the prime factors of 30.
PAULA KELLY: And so where our circles overlap, what’s going to go in there?
MICHAEL ANDERSON: Well that’s the numbers that they have in common. So if we look at 18, the first number we found was a 2. And in 30, the first prime number we found was a 2 as well, so I can put two here as something they have in common.
PAULA KELLY: So what they also got in common, you’ve got 2 3’s and 18, but just 1 in 30.
MICHAEL ANDERSON: So because they’ve each got at least one 3, I’m going to write a three in here. Now 30 has that five but 18 doesn’t. And 18 has that extra three, which 30 doesn’t. So that’s going to complete my Venn diagram.
PAULA KELLY: OK. Still looking for our highest common factor.
MICHAEL ANDERSON: Yeah. So this is the key to helping us work them out. So with our highest common factor, all we have to do is multiply the numbers in this overlap. They’ve got a 2 and a 3 in common, and 2 multiplied by 3 gives us a highest common factor. So h and cf of 6.
PAULA KELLY: OK, and then what else can we as a Venn diagram for?
MICHAEL ANDERSON: Well we’ve got all of the prime factors of both 18 and 30. So to find the lowest common multiple, all we have to do is multiply all of these numbers together. So 5 multiplied by 2 is 10, multiplied by 3 is 30, multiplied by 3 again gives us 90.
PAULA KELLY: So if we written out our 18 times table and our 30 times table, the first number in both lists would be 90. And if we draw and in some factor trees for 30 and 18, the highest number in both first lessons would be 6.
MICHAEL ANDERSON: Yeah. So 18 and 30 have lots of common factors. They’ve got 1, they’ve got 2, and they’ve got 3, but the largest number you could find is six.
In an earlier step, we saw how the highest common factor of two numbers and the lowest common multiple of two numbers can be found by listing the factors or the multiples of each number. This method works well for small numbers but is less efficient when dealing with larger numbers.
In this video, Paula and Michael show an alternate method of finding the lowest common multiple and the highest common factor using prime factor trees and a Venn diagram.

Work out and discuss

Try both methods to find the highest common factor (HCF) and lowest common multiple (LCM) of three numbers; 12, 20 and 84.
Consider the two different methods of finding the HCF and the LCM.
What do you consider the advantages and disadvantages of each method when considering three numbers? Which method would you teach to your students? Would you teach both methods?

Teaching resources

This MEP resource from CIMT covers: factors and prime numbers, prime factors, index notation, highest common factor, lowest common multiple, squares, and square roots. The initial file forms part of the textbook. The activities sheet, extra exercises and mental tests complement the work covered in the textbook. See also two videos showing further examples of finding HCF and LCM for two numbers.

Weekly problems

Now complete question 10 from this week’s worksheet.
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