Prime factors are a beautifully elegant concept which can help you with a variety of advanced 11+ (and 13+ maths questions). It also crops up at GCSE level, and indirectly at A-level. We're going to explain the basics, and then show you a few typical applications.
Question: Express 196 as the product of its prime factors
I've seen this exact question (with different numbers!) come up at 11+, 13+ and GCSE level. How do we tackle it?
Step 1: Divide 196 by any of its factors (prime or otherwise).
In this case, we've chosen 4. 196 ÷ 4 = 49, so we get this:
Step 2: If the factors that you broke 196 down into are not primes, continue breaking them down until you get to primes.
Once you get primes, circle them. Primes, by definition, will not break down any further.
Step 3: All circled numbers are prime factors of 196. So 196 = 2x2x7x7.
NB: I doesn't matter which factors you choose initially, you'll always get down to the same prime factors.
1. Sneaky problem solving
This question is from 2014 G2 London Girl's School Consortium 11+ Maths Paper
Here, you can quickly find out that 105 = 3x5x7. That means B, E and T are 3, 5, 7 (although you don't know which is which). By comparing with the other words, you can see that B = 5, and go on from there. The question is doable without knowing about prime factors, but doing the tree makes it all much easier!
Use prime factors to find pairs of numbers which multiply to give you 168. Then narrow it down by finding the pair which adds to give 34.
Similar to the previous question. You find the prime factors of 2385, and then use them to generate pairs of numbers which multiply to 2385. You then look for a pair where both numbers are between 40 and 60.
2. Finding the Square Root of large, unfamiliar square numbers
For the 11+ and 13+ maths exams, it's useful to memorise the first 15 square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. Students need to be able to recognise these numbers instantly, because they work their way into a variety of questions. For example:
A square has an area of 144cm², what is its perimeter?
But aside from memorising, there's also a really neat way to square root using prime factors. I've chosen 196 for ease, but it will work for any square number.
Step 1: Express the number as the product of its prime factors.
So 196 = 2x2x7x7
Step 2: Divide the prime factors into two equal brackets:
196 = (2x7) x (2x7)
Step 3: Simplify the brackets
196 = 14 x 14
So √196 = 14
Pretty straightforward, but a great way to cement your understanding of primes, squares, and the number system in general. I've also seen the question come up directly on a St Paul's Girls' School 11+ paper, and indirectly a number of times on other 11+ and 13+ papers. It's the kind of concept that would be particularly at home on an Eton scholarship 13+ paper.
NB: If you try to square root a non-square number, you simply won't be able to divide the prime factors into two equal brackets. For example, 8 = 2x2x2. There's no way to split 3 numbers evenly into two brackets, so we can see that 8 is not a square number.
Prime factors are a handy topic to get to grips with early. At 11+ and 13+ they'll help you tackle some of the really tricky questions, so I particularly recommend that bright, ambitious students review this topic. Prime factors are also highly useful for those attempting pre-tests, as understanding them can do wonders for your speed.