Finding the GCF and LCM of monomials

Question

The Facts: 

A hardware store has two pieces of glass. The area of the first piece is represented by the monomial 36x^2y^3. The area of the second piece is represented by the monomial 48x^3y^5.

 

Question A: 

Both pieces of glass have the same length. What is the largest possible length? (Explain) 

 

Question B: 

What are the widths of two pieces of glass? (Explain) 

 

Question C: 

What is the least common multiple of the widths of the glass? 

 

 

 

Thank you! This is the only problem I'm stuck on in my study guide review! 

Answer 1

Well, to find the GCF (Greatest Common Factor) and the LCM (Least Common Multiple) [which uses factoring to find], it makes sense to factor the two expressions (the two monomials).  Let’s do that (and we want Prime Factors):

  36x²y³    has factors:   2*2*3*3*x*x*y*y*y

  48x³y⁵   has factors:   2*2*2*2*3*x*x*x*y*y*y*y*y

 

Finding the GCF is simply a matter of collecting the common (there’s that word!) factors.  Here, they are:      2*2*3*x*x*y*y*y   =  12x²y³

 

Finding the LCM means finding the value that includes all of the factors of the two expressions (with the least number of them possible – there’s that word “least”).  Here, it is:

             2*2*2*2*3*3*x*x*x*y*y*y*y*y  =  144x³y ⁵       (both values divide into this)

 

Questions A: The largest possible common length (to have the same length and to get the area of the given expression) is the GCF = 12x²y³.

 

Question B:

               That means the widths are (all the remaining factors):

                              3             because    3 * 12x²y³ = 36x²y³

                  And   2*2*x*y*y = 4xy²    because   4xy² * 12x²y³ = 48x³y⁵    

                                                                                                 (the original expressions for areas)

 

Question C:  (see above)

      LCM =  144x³y⁵     

Answer 2

Chloe for this problem, try to think of the general equation for area of rectangles to determine the lengths and widths. The answers will be a little out of order.

 

You know that the first piece has an area of 36x2y3 and the second with 48x3y5. You also know that they have the same lengths but different widths. Because they have a rectangular shape and the equation is l•w, you can divide them to find the sides.

 

Let's set it up: have A1= 36x2y3 and A2= 48x3y5.

Since the lengths are the same, we can use l for both. W1 can be for A1's width and W2 for A2's width. The division would look like this:

 

A1/A2 = (l•w₁)/(1•w₂)

 

Because l appears in the numerator and denominator, it cancels out leaving you with w1/w2. This is key! It's telling you that if you divide the areas by each other, you will get each area's width. So let's solve:

                  **hint for length later on: 36 and 48 are divisible by 12**

36x2y3        36 • x2 • y3          3              3

--------   =   ------------    =   -------   =   ------

48x3y5        48 • x3 • y5      4•x•y2         4xy2

 

Because it was set up as A1/A2, we know that the numerator is w1 and the denominator is w2. Thus the width of A1 is 3 and the width of A2 is 4xy2.

 

In order to find each area's length, we must find out what was removed from both areas in other words the common factor! So what is the greatest common factor for both widths so that when multiplied, it will equal their respective area's? For each variable, what was taken from both the numerator and denominator of both area's?

 

I hinted earlier that 36 and 48 are both divisible by 12 so that is part of it since that is the greatest common factor of both numbers. In both widths, x2 and y3 were taken out so all together it means that 12x2y3 was the common factor and therefore the length!

 

For the least common multiple, think of what is the lowest value among the variables that would be possible to remove from.