Highest common factors explanation?

[Ludoole]

Does anyone know a simple way to explain to my year 7 ds how to find the highest common factors and lowest common multiples of 256 and 16?
I am completely clueless

Highest Common factor is the biggest number that divides into both of them.
The proper way to find it is to work out the prime factors of each number and then take the ones that appear in both lists.
So
16 = 2x2x2x2
256 = 2x2x2x2x2x2x2x2
So the HCF is in fact 16.

The LCM is the smallest number that both of them go into.
Which is the 'union' of the prime factors, which in this case is 256.

Say 56 and 42.

56 = 2x2x2x7
42 = 2x3x7

So HCF is 2x3=14 and LCM is 2x2x2x3x7=168

Oh, and to find the prime factors you can keep splitting into factor pairs until you get to the prime numbers
So 1014 = 6 x 169 = 2x3 x 13x13

Thanks Tween and teen. Hopefully i can explain that to ds in the morning

I work it out slightly differently
For HCF, I would write a list of the factors- i call them factor ladders because they have to be in pairs
Eg HCF of 12 and 20
12
1x12
2x6
3x4

20
1x20
2x10
4x5

4 is the highest number that is in both ladders. Not as advanced as other answer but might be an easier way to start off.

Similarly, I would write out multiples of the two numbers.
Eg LCM of 10 and 25

10, 20,30, 40,50,60,70,80,90,100

25,50,75,100

100 is the first to appear in both lists of multiples

It is essential for a Y7 student to understand the difference between a factor and a multiple.

I have found that whilst a child may appear to comprehend the difference, they can easily get them mixed up a week or so after they have been working on them.

likklemum yes that would work fine and is, now you come to mention it, how my DD1 was first taught.

My DD often finds the 'simplified' methods a bit confusing so I often teach her a different, supposedly 'more advanced' method that she can understand.

So OP you could try either method and see what works best for your DC.

OP look away now.

TeenandTween Is that how it's taught in schools? Having to factorise both the numbers is OK for toy examples, but prime factorisation is not something you want to be doing casually (to grossly dumb down a very complex question, there are no "fast" algorithms for doing it). I can see that's a way to explain what it means, but would that be shown as a practical method?

In real applications, you'd calculate the GCD using the Euclidian algorithm, and the LCM by multiplying the two numbers together and dividing by the GCD. And it seems a shame to miss talking about the Euclidian algorithm, because it's wonderful and beautiful and everywhere, including every time anyone logs onto Mumsnet. And if you're teaching programming, it's a lovely opportunity to explain recursion, too.

en.wikipedia.org/wiki/Euclidean_algorithm

In secondary school to teach HCF for a low pair of numbers such as 15 and 20 we would list the factor pairs of each number and find the highest number on both lists.

For a much larger number like in the OP they would be expected to use the prime factorisation of the numbers as Teen describes. An easy way to visualise this is by putting the prime factors of each number into a Venn diagram and using the union and intersect. Prime factorisation is usually achieved using factor trees (sometimes called cherry trees).

Euclid's algorithm isn't taught.

likklemum According to your explanation wouldn't 50 be the LCM of 10 and 25?

duhgldiuhfdsli My DD was taught the way noble says. noble is a maths teacher, I'm just a parent with a maths degree (who can't remember Euclid's algorithm).

Gun yes likkle missed spotting the 50; 50 is the LCM of 10 and 25.

"For a much larger number like in the OP they would be expected to use the prime factorisation of the numbers"

OK, thanks.

Shame: it's also an opportunity lost to talk about "hard" problems, which computers struggle with. Like integer factorisation.

Hi Ludole:

I tend to find that with 'new' maths terms or terms I can't quite remember definitions for myself - the easiest thing is the type in the term and see what videos pop up.

I found this one which goes through both terms: Highest Common factor/ Lowest Common multiple step by step:

BBC Bitesize KS3 - is always a great source of review/ information. Just typing in BBC Bitesize Highest Common Factor I went straight to this: www.bbc.co.uk/schools/gcsebitesize/maths/number/primefactorshirev1.shtml & within that there was a link to their pages on factors - which explains highest common factor & least common multiple here: www.bbc.co.uk/schools/gcsebitesize/maths/number/factorsmultiplesrev1.shtml

HTH

Oops sorry!