I am a former maths teacher and owner of DoingMaths. I love writing about maths, its applications, and fun mathematical facts.
What Are Lowest Common Multiples and Highest Common Factors?
Before we look at how to find the lowest common multiple and highest common factor of two numbers, we should first look at exactly what they are.
The lowest common multiple (also known as the least common multiple) of two numbers is, as the name suggests, the smallest number which is a multiple of both of the original numbers.
For example, to find the lowest common multiple of 2 and 3, we could list the first few multiples of each:
2 - 2, 4, 6, 8, 10, ...
3 - 3, 6, 9, 12, 15, ...
We can see that the smallest number that appears in both lists is 6, hence the lowest common multiple of 2 and 3, LCM(2, 3) = 6.
The highest common factor (also known as the greatest common factor) of two numbers is the largest number which divides perfectly into both of the original numbers without leaving a remainder.
For example, to find the highest common factor of 9 and 12, we could list the factors of each:
9 - 1, 3, 9
12 - 1, 2, 3, 4, 6, 12
The largest number that appears in both lists of factors is 3, hence the highest common factor of 9 and 12, HCF(9,12) = 3.
Finding the LCM and HCF in More Difficult Examples
With the numbers used above, it was relatively easy to find the LCM and HCF by listing multiples and factors and comparing the lists. This is great for numbers with relatively few factors or whose LCM comes quite early in their respective times tables.
If we wanted to find the LCM and HCF of a larger pair of numbers, which may have more factors and trickier times tables, we still have a quick and efficient method.
In order to find the LCM and HCF, we will use the fact that all numbers can be rewritten as the product of their prime factors. If you are unsure of how to do this, take a look at my article on 'How to Write a Number as a Product of Its Prime Factors'.
To see how the prime factors can be used, we will start off with a simple example.
Finding the LCM and HCF of 24 and 30
By using a factor tree, we can rewrite 24 and 30 as the products of their prime factors.
24 = 23 × 3
30 = 2 × 3 × 5
We are now going to put these factors into a Venn diagram which can be seen below. Both 24 and 30 have 2 and 3 as prime factors, hence we put 2 and 3 into the shared area in the centre of the Venn diagram. We now complete the rest of the diagram with the remaining factors.
24 has an extra two 2s in its prime factorisation to go with the 2 and 3 already used, hence we put these two 2s on the left. 30 just has the 5 remaining and so we put this on the right.
You can now see that the oval for each of our original numbers contains all of its prime factors, with a 2 and a 3 being shared between them.
The Prime Factors of 24 and 30 in a Venn diagram
Using the Venn Diagram to find the HCF and LCM of 24 and 30
Now we have our prime factors in the Venn diagram, the rest is fairly simple.
Firstly, let's find the HCF. To be a common factor of both 24 and 30, a number must only contain prime factors shared by 24 and 30. It follows from this that to be the highest common factor, it must have all of the shared prime factors.
To find the HCF we therefore multiply all of the shared prime factors from the centre of our Venn diagram. In this case we get HCF(24, 30) = 2 × 3 = 6.
Secondly, we will find the LCM. To be a multiple of 24, our number must contain all of the prime factors of 24. Likewise it must contain all of the prime factors of 30. To find the LCM we therefore multiply together all of the prime factors within our Venn diagram getting LCM(24, 30) = 2 × 2 × 2 × 3 × 5 = 120.
A Trickier Example - HCF and LCM of 120 and 252
Let's look at an example with even larger numbers now. If we wanted to find the lowest common multiple and highest common factor of 120 and 252, we would be here a long time if we tried to do it by writing out all of the factors and times tables until we found a match. For these two numbers, the prime factorisation method is particularly useful.
Our first step, as before, is to write out each number as the product of prime factors.
120 = 23 × 3 × 5
252 = 22 × 32 × 7
We then place these factors into our Venn diagram, noting that 120 and 252 share two 2s and one 3 which must go in the central section.
Prime Factors of 120 and 252 in a Venn Diagram
Finding the HCF and LCM
As before, we use the shared factors for the HCF and all of the factors for the LCM.
HCF = 2 × 2 × 3 = 12
LCM = 2 × 2 × 2 × 3 × 3 × 5 × 7 = 2520
I'm sure you can see how long it would have taken to get to 2520 if we had written out the 120 and 252 times tables!
So to find the HCF and LCM of any pair of numbers, we must first rewrite each number as the product of its prime factors and then place these into a Venn diagram.
To find the HCF we then find the product of the shared factors from the centre of the diagram.
To find the LCM, we find the product of all of the factors in the whole diagram.
This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.
© 2021 David