## Primed

Recently I’ve seen a couple of things on twitter that I’ve thought quite interesting because they have got me thinking about the way I tackled them.the first was this:

It was shared by John Rowe (@MrJohnRowe) and posed a nice question looking at two cuboids that had the same volume asking for the side lengths. It said to use digits from 0-9 without repetition. My immediate thought was to consider prime factors:

2, 3, 2^2, 5, 2×3, 7, 2^3, 3^2

I only had one each of 5 and 7 so discounted them. And I had an odd number of 2s. This meant I’d need to discount either the 2 or the 8. Discounting the 2 left me with factors of 2^6 and 3^4 thus I needed 2^3 x 3^2. I had a 1 I could use too so I had 1x8x9 = 3x4x6. This seemed a nice solution. I also considered the case for discounting the 8. I’d be left with 2^4 and 3^4 so would need products of 2^2×3^2. 1x4x9 = 2×3×6. Also a nice solution.

Afterwards I wondered if John had meant I needed to use all the digits, I started thinking about how this could be done and realised there would be a vast amount of possibilities. I intend to give this version of the the problem more thought later.

The next thing I saw was this:

From La Salle Education (@LaSalleEd). Again it got me thinking. What would I need to get the product 84? The prime factors I’d need would be 2^2x3x7. This gave me only a few options to consider. 1 cant be a number as the sum will always be greater than 14 if 1 is a number. 7 has to be one of the numbers, as if its multipled by any of the other prime factors we have already got to or surpassed 14. So it could be 2,6,7 (sum 15) or 4,3,7 (sum 14) so this is our winner.

These problems both got me thinking about how useful prime factors can be, and they both have given me additional thoughts as to what else I could and should be including in my teaching of prime factors to give a deeper and richer experience.

Prime factors can crop up so many places, and I feel sometimes people let them get forgotten or taught in isolation with no links elsewhere. I always use them i lessons o surds and when factorising quadratics but know lots don’t do this.

*I will write more on prime factors later, but for now I’m tired and need sleep. If you have any thoughts on prime factors or any additional uses not mentioned here I would love to here them. Please let me k ow i the comments or on social media.*

N.b. The La Salle tweet includes this link which takes you to a page where they are offering some great free resources.

## Equal Products

I come across a lot of puzzles and other maths things online and often save them for later, this evening I came across this little puzzle:

*The numbers 2,3,12,14,15,20 and 21 may be divided into two sets so that the product of the numbers in each set is equal. What is that product?*

I had saved this over a year ago, and cannot remember where I got it from, but I can see why. It’s a lovely little question that I intend to use as a starter next week and see how my classes get on.

**How I approached it**

Before you read on have a go at it yourself. Go on, you know you want to……..

Right, good, now you can see if I went about it the same way!

My first thought was that all the fun could be taken out of this by using a calculator, typing all the numbers in and pressing square root. So when I set it I will be adding the line “and which numbers are in each set”, and this is what I set out to find.

Firstly I set out the numbers in terms of their prime factors:

Then I tallied up the prime factors:

From this I knew that the product must be 2x2x2x3x3x5x7 which is 2520.

This, of course, answers the original question but I wanted each set. I looked at the numbers and the first think I noticed was tgat 14 and 21 had to be in separate sets, as they had 7 as a factor. I also needed to split 15 and 20, my intuition suggested that 20 and 21 should be in separate boxes, but it was easy to spot that the 2s and 3s, wouldn’t work out so I placed them together and fit the rest on around them.

A nice little puzzle, I wonder how my classes will find it.

## Mersenne and his primes

On Thursday my further maths AS class and I arrived at the classroom to discover an interesting slide still displayed on the board from a previous lesson.

My colleague had been teaching a lesson on prime numbers to his year 9 class and the slide in question was about finding new primes, how much money you can earn if you do, why this is and the “Great Internet Mersenne Primes Search” (and its unfortunate acronym).

A discussion ensued about cryptography and the uses of primes. It then moved onto the mathematical monk himself and his work in number theory. In particular that he noticed that numbers of the form (2^p)-1, where p is a prime, are usually prime. These Mersenne primes have fascinated me for years. How comes so many of them are primes? Why aren’t the all?!

The class were equally fascinated and we had a great discussion. We also managed to link it to a discussion we had had the previous lesson about p vs np, as trying to factorise (2^11)-1 is fairly difficult, but it is really easy to check if 23 is a factor. The class wondered if they could set a computer to test massive numbers for prime factors. I explained that yes, you could, but it would take so long to check the massive numbers it would be worthless. So if they can find a way to do it quickly they could become very rich.

We lost around twenty minutes of matrices time, but we have plenty of time to make it up. I think all pupils left with a deeper and broader mathematical knowledge and a healthy thirst to know more- which is at least as important.