How Can You Solve for Non-Integral Solutions of the Greatest Integer Function?

[sadhu]

can anyone tell me how to solve for integer solutions of

[x]*[y]=x+y

tell the interval of its non integral solutions

pleazzzzzzzzzzzzzzzzzz...

 

[CRGreathouse]

But [x] = x for all integers x, so the integer solutions of [x][y] = x + y are the same as the integer solutions of xy = x + y, (x, y) in {(0, 0), (2, 2)}.

 

[sadhu]

well can you show the step

the answer is (0,0) ,(2,2)

i am looking for good and proper stepwise answer and not guesses , i think my method is very weak ..thats why i am here

 

[HallsofIvy]

first, I must say that "Guessing" (and then checking your guess) is a perfectly "good and proper" method! For n an integer, [n]= n so your equation is simply xy= x+ y. You can write this as xy- y= (x-1)y= x or, if x is not 1, y= x/(x-1). If x is not 0, that says x-1 divides x. The only integer x such that x-1 is a factor of x, is x= 2. You can then check x= 1 or x= 0 separately: If x= 1, xy= x+ y becomes y= 1+ y which is never true.; If x= 0, then xy= x+ y becomes 0= 0+ y which is true for y= 0. The only only solutions are x= y= 0 and x= y= 2.

 

[sadhu]

i agree to what you said

if you replace x-1=k
y=1+1/k
case 1
k>=1

y=2
no further value of k will do as 1/k is a fraction and goes on to decrease

similiarly you can do k<1

k=-1
y=0

rest all give fraction of decreasing value

ok this much is clear to me but i can't even think of something to start with for next part