What Are Factors Of Numbers - Easily Explained
Finding The Factors
Factors, factors everywhere and yet, they are such tricky little things, at least, according to young people they are!
As a mathematician, I teach a diverse range of topics involving number, algebra, geometry and statistics every day and even after 15 years it never ceases to surprise me how many pupils, regardless of age and ability, struggle with a simple question such as "List all the factors of 40"
These same pupils will readily solve equations, unravel complex problems involving geometric reasoning and perform advanced statistical calculations and yet, when it comes to listing the factors of a number such as 40, they suddenyl seem to lose all mathematical ability.
So, let me address the troublesome question of factors.
Factors are integers (whole numbers - the numbers we use to count with) that go into other numbers without a remainder.
There. Seems simple enough, doesn't it?
And yet, in answer to the question about listing the factors of 40, pupils will write all sorts of things. Quite commonly they will mistake factors for multiples and so, as their answer, they will write:
40, 80, 120, 160, 200 . . . . .
When we are looking for the factors of a number such as 40, what we are looking for are all the numbers that will 'go into' 40.
Various people I have met and taught with have their own way of explaining factors and their own ways of explaining pupils how to work them out.
For my part, I prefer the simplistic and systematic 'listing' approach, like this:
Let's find out all the ways we can 'make' 40 by multiplying:
1 x 40
2 x 20
4 x 10
5 x 8
Our next step would be 8 x 5 but as we have already used 8 and 5, we stop because otherwise we will be repeating factors we already have.
So, by using this simple and systematic 'listing' approach, we can easily see that the factors of 40 are:
1, 2, 4, 5, 8, 10, 20, 40
Highest Common Factor (HCF)
Now, if factors weren't difficult enough, we also have a task where we have to find the highest common factor (HCF) of two numbers.
"Find the HCF of 40 and 64"
Tricky little activity!
To perform this mathematical masterpiece, we need to find all the factors of 40 and of 64 and put them into 2 lists:
40 = 1, 2, 4, 5, 8, 10, 20, 40
64 = 1, 2, 4, 8, 16, 32, 64
To find the HCF of 40 and 64 is now easy. Just look for the biggest number that appears in both lists.
In this case, the largest number I can find is 8, so the HCF of 40 and 64 is 8
There . . . . not too difficult really, is it?
Now we come to the trickiest part of all, Prime Factors.
Prime factors are simply prime numbers that multiply together to make a number.
For instance, the prime factors of 40 are all the prime numbers that we can multiply together to make 40:
I teach this by using a factor tree
40 = 5 x 8
8 = 4 x 2
4 = 2 x 2
The numbers underlined and in bold are the prime numbers that will 'go into' 40 so we call them prime factors.
The prime factors of 40 are 2 x 2 x 2 x 5 or 23 x 5
Stay tuned for the next simple maths topic.
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