find the greatest common factor of these two expressions

Question

8x^7 w^8 y^5 and 28x^2 y^3

Answer 1

8x⁷w⁸y⁵

28x²y³

  

Now, you will soon be looking for prime factors and subtracting exponents to find the GCF.  Just to make it very clear, let's write each expression out and find what's common:

    2*2*2*x*x*x*x*x*x*x*w*w*w*w*w*w*w*w*y*y*y*y*y

    2*2*7*x*x*y*y*y

 

O.K., either highlight, circle, or move the common items to the front (let's do that):

   2*2*x*x*y*y*y  *  2*x*x*x*x*x*w*w*w*w*w*w*w*w*y*y*

   2*2*x*x*y*y*y  *  7

 

So, the GCF is 4x²y³

Answer 2

Hi Emma, the greatest common factor would be the largest integer that occurs in both 8 and 28, the highest power of x, y, and w that occurs in both expressions.

 

Greatest common integer:

 

Greatest common integer in 8 and 28.

 

One way to do this is to find the prime factorization of both 8 and 28.

 

Prime factorization of 8: 2 * 2 * 2

 

Prime factorization of 28: 2 * 2 * 7

 

The two's occur 3 times in 8 and 2 times in 28.

 

The greatest common integer is 2 * 2 or 4.

 

A second way to find the greatest common factor of both 8 and 28 is by looking for integers that divide both.

 

2 will divide both 8 and 28.

8 / 2 = 4

 

28 / 2 = 14

 

2 will again divide both 4 and 14

 

4 / 2 = 2

 

14 / 2 = 7

 

Since both 2 and 7 are prime numbers, there is no common divisor.

 

The greatest common factor is 2 * 2 or 4.

 

Powers of x:

 

The greatest power of x will be the largest power that occurs in both x⁷ and x². That is 2

 

So the greatest power of x is 2.

 

Powers of w:

 

The greatest power of w will be omitted from the greatest common factor because only one of the expressions has a w in it.

 

Powers of y:

 

The greatest power of y will be the largest power that occurs in both y⁵ and y³. That is 3.

 

So the greatest power of y is 3.

 

The greatest common factor:

 

The greatest common factor is: 4x²y³.

 

Questions?