What is a Prime Factor? Something like biting into the sweet layers of a Backlava

These. Two. Words.

Prime Factor.

Seem to have the power to completely muddle and befuddle and sometimes horrify my students – ok I’m being dramatic (it’s halloween), but so many students find them confusing and actually have no idea what a prime factor is.

Panic ensues if this question comes up in an exam.

I think the panic is worse because you all know that it’s NOT difficult…its an easy thing right? But then why is it so hard to know what to do?

Let’s look at a typical question:

Write 20 as a product of prime factors. Give your answer in index form.

(By the way this is from the fat textbook: CGP AQA Maths Higher Level, which you can buy from the shop here.)

So let’s read that again:

Write 20 as a product of prime factors. Give your answer in index form.

What is product in maths? Multiplying. So we could say write 20 as a multiplying of prime factors…

What is a prime number?

A number that only divides by itself and 1. Here are some prime number examples:

2, 3, 5, 7, 11, 13, 17, 19, 23

Those numbers hardly appear in any multiplication tables.

They only divide by themselves and 1, which means they are only in their own list of times tables.

i.e. 1×3 = 3 – which other multiplication sum has an answer of 3?

Ok, so we know what a prime number is now.

What is a factor?

A factor is a number that divides exactly into another number.

E.g. the factors of 24 are: 1, 24, 2, 12, 3, 8, 4 and 6.

Those are all of the numbers that you can divide or share 24 by.

An easy way to find the factors of a number is to use a layered tree diagram:

(picture coming soon)

A prime number only has 2 factors, itself and 1.

Here are some prime numbers and their factors:

13 – Divides by 1 and 13

17 – 17, 1

23 – 23, 1

What is a prime factor?

Let’s have another look at the factors of 24:

1, 24

2, 12

3, 8

4, 6

Are any of those factors also prime numbers?

Yes, the 2 and the 3.

2 and 3 are both prime numbers and they are also factors of 24.

So the 2 and 3 are prime factors of 24.

So a prime factor is a number that divides into another number but is a prime number itself.

Let’s look again at the question:

Write 20 as a product of prime factors. Give your answer in index form.

We might not fully understand that just yet. But we can have a go with the prime factors.

Let’s list the factors of 20:

1, 20

2, 10,

4, 5

The 5 and the 2 are prime factors. The rest are not.

We know product means to multiply. So what if we multiply our two prime factors?

\[2\ X\ 5\ =\ 10\]

Hmmm, that’s not quite 20.

Let’s have another look at the factor pairs.

We know that 4 times 5 is 20.

What if we break down the 4 into 2 times 2?

That would mean we are listing all of the factors as prime factors.

20 = 4 x 5 = 2 x 2 x 5

We could also do it with the other factor pairs

20 = 2 x 10 = 2 x 2 x 5

So we’ve done part one of the question, we have expressed or written 20 as a product of its prime factors.

What exactly does index form mean?

Index is another word for powers in maths – that little number that floats at the top.

So If we look again at the numbers, we see we have 2 x 2 which can be rewritten as 2 sqaured:

\[20\ =\ 2^2\ x\ 5\]

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