Intregal solution

Reason:
x and y both can't be even or both odd(since sum of two even or two odd numbers is always even)
So even if at all a solution exists one has to be even and one has to be odd.
let x=2k
y=2r+1
x²+y²=4k²+(2r+1)² which is 4(k²+r²+r)+1 or of the form 4p+1
since 2007 is not of the above form,no solutions.

p.s:here r,k,p all are integers...

this question was already discussed, ans is (A)

I took an A4 size paper to solve it.