Find the number of integer solutions to $x^2 + y^2 = 2007$.
Now, $x^2 + y^2 = 2007 =$ odd. One of $x$ and $y$ is even and another is odd. Let $x$ is even and $y$ is odd. Now, $x^2 \equiv 0 \pmod{4}$ and $y^2 \equiv 1 \pmod{4}$. Dividing the equation by 4: $0 + 1 \equiv 3 \pmod{4}$, i.e. $1 \equiv 3 \pmod{4}$, which is impossible. No solution. (A)
Find the number of integer solutions to $x^2 + y^2 = 2007$.