解取整方程

y = 720(x - [x]) . . . . . . . . . . . (i)
1/4 + x/720 = y - [y]
==> 180 + x = 720(y - [y]) . . .(ii)
From (i), (ii), we know that the value of x is at least -180, the value of y is at least 0. Assume the solution of x is between a and (a + 1), y is between b and (b + 1), then from (i), (ii), we get :
y = 720x - 720a . . . . . . . . . . (iii)
180 + x = 720y - 720b . . . . . (iv)
Solve (iii), (iv), we get,
x = (518400a + 720b + 180)/518399
y = (720a + 518400b + 129600)/518399
Check : When a = b = 0, then one of the solution (x, y) is
(180/518399, 129600/518399), which is ok.
When a = 10, b = 20, then one of the solution (x, y) is
(5198580/518399, 10504800/518399), which is also ok.
Ans : For all a >= -180, b >= 0,
(x, y) = ((518400a + 720b + 180)/518399, (720a + 518400b + 129600)/518399)





2012-04-24 10:31:08 補充：
One of the integral solution is (x, y) = (-180, 0)

2012-04-25 20:18:18 補充：
自由自在 :
Thanks so much, I forget to consider the upper limit.

2012-04-25 20:38:39 補充：
I would like to express my sincere thanks to 自由自在( 知識長 ),

The answer should be :
For all -180 <= a <= 539, 0 <= b <= 719 and except (a, b) = (539, 719),
(x, y) = ((518400a + 720b + 180)/518399, (720a + 518400b + 129600)/518399)

There are all together 720*720 - 1 = 518399 solutions only.

mfmartinfan:
it should be noted that -180<=a<=539 and 0<=b<=719
And for a=539 and b=719 the calculated results is rejected
So there are all together 720x720-1=518399 solutions only.

我唔知咁樣岩唔岩...
[x]=x-1
so y=720{x-(x-1)
y=720
[y]=720-1=719

1/4+x/720=1
x/720=3/4
x=540

so [x]=539