The GCF of two numbers p and q is 5. Can you find the GCF of 6p and 6q? Explain your answer.

Question

This is one of my questions on my homework that I just cannot figure out. 

Answer 1

The Greatest Common Factor (GCF) is the largest (greatest) factor (a factor multiplied by another factor produces a term) that is common to two or more values.

 

So, if a problem asks for the GCF of 12 and 6, we first factor:

    12  =  1 * 2 * 2* 3

      6  =  1 * 2 * 3

Then, we conclude that 1*2*3 = 6 is the GCF.

 

Now, we could write absolutely all of the possible factors:

   For 12:   1, 2, 3, 6, 12

   For 6:     1, 2, 3, 6

 and choose the largest number that is shared by the two lists.

 

However, writing the product ("*") of prime factors allows us to have a list that can be grouped when finding all the common values.

 

Here is a very important point:

   If the GCF is 5, then p and q can be written like this:

      p  =    1*5*a       [where a is some number]

      q  =    1*5*b       [where b is some other number]

 

So, what is the GCF of:

    6p  =  (6)*1*5*a = 1*2*3*5*a    

    6q  = (6)*1*5*b  = 1*2*3*5*b   

The GCF= 1*2*3*5 = 30      [note: a and b may have factors or else be prime --

        we don't know, so we can't factor them]

 

Answer 2

Alan is correct that if the greatest common factor is 5

and you increase the numbers by a factor of 6 then the

new greatest common factor is 5(6) = 30

 

You can take an example with two numbers whose GCF is 5.

Let's use 5 and 10

 

      The prime factors of 5 are simply 5  ... (and 1, which is a factor of every number)

      The prime factors of 10 are 5 * 2

      GCF = 5

   

     6(5) = 30:  Prime factors  2*3*5

     6(10) = 60: Prime factors  2*2*3*5

 

     Greatest common factor of 30 and 60 = 2*3*5 = 6*5 = 30 

                

Answer 3

Since 6p and 6q already have a common factor of 6 and the GCF of p and q is 5, the GCF of 6p and 6q is 30.