# Thread: Show that all non integral solutions lie on exactly two lines

1. ## Show that all non integral solutions lie on exactly two lines

Consider the equation $\lfloor x \rfloor \lfloor y \rfloor = x + y$.

Show that all non-integral solutions of this equation lie on exactly two lines.

2. Originally Posted by fardeen_gen
Consider the equation $\lfloor x \rfloor \lfloor y \rfloor = x + y$.

Show that all non-integral solutions of this equation lie on exactly two lines.
The left hand side is an integer whether x and y are or not. If x and y are not integer, then the "fraction parts" must cancel. Either x= a+ r, y= b- r or x= a- r, y= b+ r, where a and b are integers, 0< r< 1. Those give the two lines. (And, of course, ab= a+b.)

3. Originally Posted by HallsofIvy
Either x= a+ r, y= b- r or x= a- r, y= b+ r, where a and b are integers, 0< r< 1.
Hi HallsofIvy.

What about $x=a+r,\ y=b+s$ where $0 and $r+s=1?$

4. $x+y=0$ and $x+y=6$

5. Originally Posted by TheAbstractionist
Hi HallsofIvy.

What about $x=a+r,\ y=b+s$ where $0 and $r+s=1?$
Isnt that the same thing as HallsofIvy's suggestion? $r+s = 1 \implies y = b+(1-r) = (b+1)-r$, where b+1 is just another integer.

6. [x][y]=x+y

[x][y]=[x]+[y]+{x}+{y} ({x} and {y} denote the fractional parts of x and y respectively)

0 $\leq${x}+{y}<2

0 $\leq$([x][y]-[x]-[y])<2

1 $\leq$(1+[x][y]-[x]-[y])<3

1 $\leq$([x]-1)([y]-1)<3

([x]-1)([y]-1)=1,2

Case 1

([x]-1)([y]-1)=1

[x]-1=1 and [y]-1=1

[x]=[y]=2

{x}+{y}=(2)(2)-2-2=0

{x}={y}=0,but then $x$ and $y$ will be integers

Case 2

$([x]-1)([y]-1)=2$

$
[x]-1=-1 and [y]-1=-2
$

$[x]=0 and [y]=-1$

{x}+{y}=0-0-(-1)=1 and thus $x+y=0$

Case 3

$([x]-1)([y]-1)=1$

$[x]-1=-1 and [y]-1=-1$

$[x]=0 and [y]=0$

{x}+{y}=0-0-0=0

{x}={y}=0,but then $x$ and $y$ are integers

Case 4

$([x]-1)([y]-1)=2$

$[x]-1=1 and [y]-1=2$

$[x]=2 and [y]=3$

{x}+{y}=6-2-3=1

Therefore, $x+y=6$