N Product of Primes (GCSE Maths)
Express the following numbers: 72 & 180, as a product of prime factors.
The first thing you need to do is consider the number itself and the smallest number
that can divide into it.
You are pretty sure that the number you think of is a prime number.
Here, the number 1 does go into the number 72, but 1 is not a prime factor.
The number 2 goes into 72, so we are allowed to divide by this.
2 72
This is a simply a division sign with an upside down, so you have more space underneath to
keep calculating.
2 divided into 72 is 3 remainder 1.
2 into 12 is 6.
Here, you can divide 36 again by the number 2, two into three is one remainder one, two
into 16 is 8.
Once again, you can divide by the number 2 here.
Two into number 18 is 9.
Now here, we can no longer divide by 2.
So we think of the next number the goes into 9, which is the number 3.
Again, 3 is a good number to use because it is a prime number.
How many 3s are there in 9?
It is 3.
Once again here, you can divide the number 3 by itself.
The result would be the number 1.
The moment you see the number 1, you need to stop.
Your calculation is complete.
2 72
2 36
2 18
3 9
3 3
1
These are all the factors that divide into the number 72.
So together, these numbers create the number 72 when they are multiplied together.
The final answer here is 2 x 2 x 2 x 3 x 3.
You could leave your answer in this form, but generally, a question will ask you to
write it in index form.
This means combining any of the similar bases together and writing them in a simpler form.
So 2 x 2 x 2 can be written as 23 x 32, where 3 x 3 is the same as 32.
This is your final answer.
Let’s have a look at this number now, a 180.
Again, the smallest number that this can be divided by is the number 2.
Half of 180 is 90; divide this again by the number 2, to get the number the number 45.
This is where some people may alter their method.
2 180
2 90
45
Sometimes, you can think of the lowest number you can think of that goes into 45, which
is the number 3.
However, when most people see the number 45, you think of the number 5 being the first
prime number that can divide into it.
That’s okay, as long as you know the number you are dividing by is a prime number.
So here, I’m going to divide by the number 5.
2 180
2 90
5 45
3 9
3 3
1
I’ll do it again in a minute, where I will do the number 3 first and show you that both
of these methods give you the same answer.
So divide by 5 because it is a prime number and you get the number 9.
Now, we can divide 9 by 3, as it is also a prime number.
We get the answer 3 and once more, like before, we divide this by 3 and you get the number
1.
Again, when you see the number 1, you need to stop.
Your calculation is complete.
Let’s combine these together.
2 x 2 is equivalent of 22, multiplied by 5 that’s only one, so we can just slide the
number 5 down.
3 x 3 is the same as 32.
22 x 5 x 32
If I was to do this again using the rule of the lowest number you can think of to divide
by:
2 180
2 90
3 45
3 15
5 5
1
You divide by 2 to get 90; we divide by 2 to get 45; we divide by 3 because it’s now
an odd number; again we divide by 3, as it is the smallest number that divides into 15
and we get the number 5 and finally, the only number that we can think of that is prime
that goes into 5 is the number 5 and we get into 1.
Again, here, we can stop, we have reached the end and when you combine these together,
you get:
22 x 32 x 5
which is exactly the same as this answer, except the numbers pop out in a different
order.
[end of audio – 05:56] N Product of Primes (GSCE Maths)
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