HCF - I don't Understand

[MickyM]

Hey,

I've never had a great mind for maths and I want to change that so I've been refreshing myself from a book called Foundation Maths by Anthony Croft & Robert Davison and I'm working through a list of problems they give for working out HCF. I've been okay with all of the problems except one and upon looking at the answer, I don't see how they arrived at it unless I'm overlooking something.
The answer performs a step which I don't understand and which isn't explained in the examples that lead up to these questions.

Here are the set of numbers given in the question to work out the HCF:-

96, 120, 144

The answer they provide is this:-

96 = 2 x 2 x 2 x 2 x 2 x 3
120 = 2 x 2 x 2 x 3 x 5
144 = 2 x 2 x 2 x 2 x 3 x 3

So, HCF = 2 x 2 x 2 x 3 = 24

The book makes no mention of performing more muliplications to arrive at a HCF result so why is it 2 x 2 x 2 x 3?
I get that we're looking for the Highest common factor but the breakdown shows that 3 is the highest common factor when we look at those results... so I must be missing something?
The book gives no example to show why this is so I'm lost :/ I fail to understand or see where 2 x 2 x 2 x 3 comes from.

Maybe someone would be kind enough to help this explain? Please

Many thanks!

Mike

 

[Deleted member 4993]

Hey,

I've never had a great mind for maths and I want to change that so I've been refreshing myself from a book called Foundation Maths by Anthony Croft & Robert Davison and I'm working through a list of problems they give for working out HCF. I've been okay with all of the problems except one and upon looking at the answer, I don't see how they arrived at it unless I'm overlooking something.
The answer performs a step which I don't understand and which isn't explained in the examples that lead up to these questions.

Here are the set of numbers given in the question to work out the HCF:-

96, 120, 144

The answer they provide is this:-

96 = 2 x 2 x 2 x 2 x 2 x 3
120 = 2 x 2 x 2 x 3 x 5
144 = 2 x 2 x 2 x 2 x 3 x 3

So, HCF = 2 x 2 x 2 x 3 = 24

The book makes no mention of performing more muliplications to arrive at a HCF result so why is it 2 x 2 x 2 x 3?
I get that we're looking for the Highest common factor but the breakdown shows that 3 is the highest common factor when we look at those results... so I must be missing something?
The book gives no example to show why this is so I'm lost :/ I fail to understand or see where 2 x 2 x 2 x 3 comes from.

Maybe someone would be kind enough to help this explain? Please

Many thanks!

Mike

For a quick review, go to:

http://www.purplemath.com/modules/lcm_gcf.htm

In USA, this parameter (that you call HCF) is called GCF.

 

[JeffM]

Hey,

I've never had a great mind for maths and I want to change that so I've been refreshing myself from a book called Foundation Maths by Anthony Croft & Robert Davison and I'm working through a list of problems they give for working out HCF. I've been okay with all of the problems except one and upon looking at the answer, I don't see how they arrived at it unless I'm overlooking something.
The answer performs a step which I don't understand and which isn't explained in the examples that lead up to these questions.

Here are the set of numbers given in the question to work out the HCF:-

96, 120, 144

The answer they provide is this:-

96 = 2 x 2 x 2 x 2 x 2 x 3
120 = 2 x 2 x 2 x 3 x 5
144 = 2 x 2 x 2 x 2 x 3 x 3

So, HCF = 2 x 2 x 2 x 3 = 24

The book makes no mention of performing more muliplications to arrive at a HCF result so why is it 2 x 2 x 2 x 3?
I get that we're looking for the Highest common factor but the breakdown shows that 3 is the highest common factor when we look at those results... so I must be missing something?
The book gives no example to show why this is so I'm lost :/ I fail to understand or see where 2 x 2 x 2 x 3 comes from.

Maybe someone would be kind enough to help this explain? Please

Many thanks!

Mike

Let's start by confirming that the book's answer is correct

\(\displaystyle 2 * 2 * 2 * 3 = 24.\)

\(\displaystyle 24 * 4 = 96.\)

\(\displaystyle 24 * 5 = 120.\)

\(\displaystyle 24 * 6 = 144.\)

So there is no doubt whatsoever that 24 is a factor that is common to 96, 129, 144. With me so far?

\(\displaystyle 2 * 2 * 24 = 96.\)

\(\displaystyle 5 * 24 = 120.\)

\(\displaystyle 2 * 3 * 24 = 144.\)

There is no factor remaining after 24 that is common to ALL THREE products. 2 is common to two of the products, but not all three.

So the highest common factor among all three products is 24. Have I convinced you?

Now here is the reasoning behind the procedure shown in your book. First, we factor each number into its prime factors.

\(\displaystyle 96 = 2 * 2 * 2 * 2 * 2 * 3.\)

\(\displaystyle 120 = 2 * 2 * 2 * 3 * 5.\)

\(\displaystyle 144 = 2 * 2 * 2 * 2 * 3 * 3.\)

So the prime factors of the three products are 2, 3, and 5.

Now we determine what is the least number of times each of these prime factors is used.

What is the smallest number of 2's used? Obviously 3. What is the smallest number of 3's used? Obviously 1. What is the smallest number of 5's used? Obviously 0.

So the highest common factor = \(\displaystyle 2^3 * 3^1 * 5^0 = 8 * 3 * 1 = 24.\)

Why does this work? Even though one product has factors of 2 * 2 * 2 * 2 * 2 = 32, 32 is not a factor of all three products. But
2 * 2 * 2 = 8 is a factor of all three products. Similarly, 3 * 3 = 9 is a factor of one product, but 9 is not a factor of all three products. And 5 is a factor of only one product. So what is common to all three numbers being analyzed is three 2's, one, 3, and no 5's. When I multiply three 2's and one 3 I get the highest factor common to all three products. If I add a factor, it will not work for all three products.

Make any sense?

 

[MickyM]

Thanks both!

@JeffM:-

Yes, thank you - that last step was the process I couldn't find in the book.
Now I think I understand. It's simply a case of looking at the least number of times the prime factors appear and then adding those numbers together to find the Highest Common Factor or GFC.

Thanks!!


Mike

 

[JeffM]

Thanks both!

@JeffM:-

Yes, thank you - that last step was the process I couldn't find in the book.
Now I think I understand. It's simply a case of looking at the least number of times each prime factor appears in any one of the products being analyzed and then MULTIPLYING rhose primes that least number of times to find the Highest Common Factor or GFC.

Thanks!!


Mike

I think you have it, but I have made a few adjustments to your description.

 

[dsk2]

Mike,

I think you are doing okay there.

If the question was what is the highest common prime factor, then you are right, 3 is the correct answer. But the question was what is the highest common factor, which means, what is the biggest number (not necessarily prime, or even or odd) that is a common factor to ALL the three given numbers. The answer is 24. You cannot find any number that is greater than 24 that will be a COMMON factor to all the three numbers. Try it.

Of course, your question has been answered, but I thought I will add my 2 cents.

Cheers,
Sai.

Hey,

I've never had a great mind for maths and I want to change that so I've been refreshing myself from a book called Foundation Maths by Anthony Croft & Robert Davison and I'm working through a list of problems they give for working out HCF. I've been okay with all of the problems except one and upon looking at the answer, I don't see how they arrived at it unless I'm overlooking something.
The answer performs a step which I don't understand and which isn't explained in the examples that lead up to these questions.

Here are the set of numbers given in the question to work out the HCF:-

96, 120, 144

The answer they provide is this:-

96 = 2 x 2 x 2 x 2 x 2 x 3
120 = 2 x 2 x 2 x 3 x 5
144 = 2 x 2 x 2 x 2 x 3 x 3

So, HCF = 2 x 2 x 2 x 3 = 24

The book makes no mention of performing more muliplications to arrive at a HCF result so why is it 2 x 2 x 2 x 3?
I get that we're looking for the Highest common factor but the breakdown shows that 3 is the highest common factor when we look at those results... so I must be missing something?
The book gives no example to show why this is so I'm lost :/ I fail to understand or see where 2 x 2 x 2 x 3 comes from.

Maybe someone would be kind enough to help this explain? Please

Many thanks!

Mike