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I pick two whole numbers $x$ and $y$ between $1$ and $10$ inclusive (not necessarily distinct). My friend picks two numbers $x -4$ and $2y-1$. If the product of my friend's numbers is one greater than the product of my numbers, then what is the product of my numbers?

 

All work shown would be appreciated!

 Jan 15, 2020
edited by Guest  Jan 15, 2020
 #1
avatar+18295 
+1

If both x and y must be whole numbers, then:  (x - 4)(2y - 1)  =  xy + 1

Multiplying out     --->        2xy - x - 8y + 4  =  xy + 1

Rearranging terms     --->     2xy - xy - 8y  =  x - 3

                                                      xy - 8y  = x - 3

Factoring     --->                            y(x - 8)  =  x - 3

                                                               y  =  (x - 3)/(x - 8)

If you test the number between 1 and 10 for x, you can get these whole number answers:  (3,0), (7,-4), and (9,6).

 

Of these, only the answer  x = 9  and  y = 6  have answers between 1 and 10 for both x and y.

 Jan 15, 2020
 #2
avatar+107405 
+1

We have that

 

(x - 4) ( 2y - 1)   =   xy + 1       simplify

 

2xy - 8y - x  + 4  =  xy + 1

 

xy - 8y - x + 3   =   0

 

xy  =  x + 8y - 3

 

xy - x  =  8y -  3

 

x(y - 1)  =  8y - 3

 

             8y - 3

x =        ______

              y -  1

 

Possibilities  for  x, y

 

x           y

13        2              reject

9          6              accept

 

So  the product of your  numbers  =  9*6  =  54

 

 

cool cool cool

 Jan 15, 2020

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