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what two numbers form 8 as the greatest common factor

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2 Answers

This is a very interesting question. A short answer is:

A pair of numbers that have k as their greatest common factor must be divisible by k, then have the resulting quotients be relatively prime.

When presented as a multiple choice question, such as:

Which two numbers have 8 as their greatest common factor?

  1. 12 & 24
  2. 16 & 24
  3. 4 & 16
  4. 16 & 32

It can be solved by the menthod already stated. You can completely factor each number and then compare the the prime factors, see what they have in common, and from there determine the greatest common factor for each pair. Eventually you'll find a pair that has 8 as its greatest common factor and you found the answer.

However, here is one way that I like to approach this problem when it is presented as a multiple choice question. As you glance ahead at the lengthy explaination ahead, I'll say that if you learn this alternative method, you'll have a new way to perform these kinds of problems and you'll have a better understanding of what you're doing. In my opnion, this is without a doubt a great thing because the better you understand this, the faster it will become. Fundamentally, we are doing the same steps, but just in a slightly different order, to make things clearer once you fully understand the process.

The first step is to try to divide each number by the proposed greatest common factor, in this case 8. If we try that, we can see that neither 4 nor 12 are divisible by 8, so the first and third choices can be eliminated.

The second step is to divide each number by the proposed greatest common factor, in this case it's still 8. The reason why I like this method is that we're already doing this in the first step to check if they are divisible by 8, so I like to think of it as being free. This would leave us with 2 & 3 from our second choice and 2 & 4 from out fourth choice.

The third step is to completely factor the remaining numbers.

The fourth step is to eliminate choices where common factors still exist. Since we have already divided these numbers by the proposed greatest common factor, there should be no prime factors in common. In this example the second choice are already prime, while the fourth choice turns in to 2 & 2×2, which we now see has a factor of two in common on both sides.

Follow up example:

Which two numbers have 60 as their greatest common factor?

  1. 360 & 840
  2. 900 & 840
  3. 360 & 1260
  4. 800 & 1260

Steps 1 & 2: Divide by GCF

  1. 6 & 14
  2. 15 & 14
  3. 6  & 21
  4. 800 & 1260 800 isn't disivible by 60.

Steps 3 & 4 Factor and elminate choices with common factors.

  1. 2×3 & 2×7 This pair has 2 in common.
  2. 3×5 & 2×7 The solution!
  3. 2×3 & 3×7 This pair has 3 in common.
  4. X

DONE! :-)

By comparison, if you wanted to completely factor everything it would begin with:

  1. 23×32×5 & 23×3×5×7

or if you didn't use exponents,

  1. 2×2×2×3×3×5 & 2×2×2×3×5×7

and then you would have to group the common factors and find the GCF for each pair either by rewriting it, or by underlining it.

 

Comments

I think your explanation for the answer tothis problem is a little too complicated.  Sorry.

Sure, not a problem. It's much easier to explain this in person than it is online.

Comment

Greatest common factor (GCF) may be defined as the greatest factor that divides two numbers.

 

To find the GCF of two numbers:

 

1) List the prime factors of each number.

 

2)Multiply those factors both numbers have in common.

 

3)If there are no common prime factors, the GCF is 1.

 

For example, 

 

Prime factors of 18  are 2*3*3

 

Prime factors of 24 are  2*2*2*3

 

the common factors here are 2 and 3 so  the greatest common factor is 2*3 = 6

 

In our question,

 

we are asked what two numbers form 8 as the greatest common factor..

 

Lets see here, the two numbers we arelooking for will definitely be multiples of 8 right??

 

so lets see the table of 8

 

8*1= 8

8*2= 16

8*3=24

 

The factors for 16 are 1, 2, 4, 8,and 16

 

The factors for 24 are 1,2,3,4,6,8,12, and 24.

 

The greatest common factor for 16 and 24 is 8 right? so there you go.... the two numbers are (16, 24)

Comments

I like your explanation.  It is easier to understand.

Comment