## Discovering How to Efficiently find the Lowest Common Multiple and Highest Common Factor

The last 2 lessons with Grade 7 (Year 8), we’ve discussed prime factor decomposition using factor trees and then finding the Lowest Common Multiple, LCM, and Highest Common Factor, HCF, of two simple numbers – e.g. Find the HCF and LCM of 12 and 16. They decided that the easiest way to find the HCF is to list all of the factors and then find the highest one and used a similar listing technique to find the LCM.

At the start of the third lesson I put this question on the board:

Find the LCM of 102 and 158?

I chose the numbers so that they were just on the brink of being findable using their method but also quite frustrating in that it would take a long time.

After they’d laboriously done that, I posed this question:

Find the LCM of 516 and 1038?

As you can imagine, at this point they were internally begging for another, more efficient method to do this. So I asked:

The first two comments set it up nicely,

1) It takes too long to do this.

2) There must be a quicker way.

To these comments I responded with:

Two good points there, I wonder if there’s a more efficient way of doing this?

Then I immediately went onto Geogebra without talking and stuck in the commands lcm[516, 1038] and hcf[516, 1038] – of course the answers popped up instantly. Obvious wow’s and ooh’s ensued until someone asked how the software had done it.

At this point, I let them investigate in any way they thought appropriate, knowing that I could spend time with some of them who still have awful structures for starting an investigation. After 15 minutes, one of the students came to me with a conjecture:

If you multiply both of the numbers and then divide by the HCF, you’ll get the LCM.

Brilliant stuff!

Of course he identified that he still didn’t know how to find the HCF efficiently. So, in the interest of keeping a good pace to the lesson and helping out those who were struggling, I decided at this point to provide more structure by giving out this sheet.

It wasn’t at all long before some of the students identified that to find the HCF, you simply multiply the prime factors that the two numbers have in common.

When we came to discuss it, the students were absolutely happy with why both of these methods worked. Much happier with this than the usual Venn Diagram technique with students struggling to remember which numbers to multiply.

At the end, just to shake it up, I showed them Euclid’s Algorithm to find the HCF of two numbers.

Find the HCF of 120 and 45.

Euclid – HCF(120, 45) → HCF(120-45, 45) → HCF (75, 45) → HCF (75-45, 45) → HCF (30, 45) → HCF (30, 45-30) → HCF (30, 15) → HCF (30 – 15, 15) → HCF (15, 15).

Hence the HCF of 120 and 45 is equal to 15.

They all loved it – but the best part is that they want to know why this works – great that they’re starting to think like mathematicians!!!