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Greatest Integer function

  1. Jan 2, 2008 #1
    can any one tell me how to solve for integer solutions of

    [x]*[y]=x+y

    tell the interval of its non integral solutions

    pleazzzzzzzzzzzzzzzzzz....
     
  2. jcsd
  3. Jan 2, 2008 #2

    CRGreathouse

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    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)}.
     
  4. Jan 2, 2008 #3
    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
     
  5. Jan 2, 2008 #4

    HallsofIvy

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    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.
     
  6. Jan 2, 2008 #5
    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 cant even think of something to start with for next part
     
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