$\lfloor 2x \rfloor \lfloor 3x \rfloor$ From $1$ to $10000$ including both, how many of those integers can be written as:
$$\lfloor 2x \rfloor \lfloor 3x \rfloor$$
Where $x$ is a real number?

To help you get you started: 
If $n \le x < n+\dfrac{1}{3}$ for some integer $n$, then $\lfloor 2x \rfloor \lfloor 3x \rfloor = 2n \cdot 3n = 6n^2$.
If $n+\dfrac{1}{3} \le x < n+\dfrac{1}{2}$ for some integer $n$, then $\lfloor 2x \rfloor \lfloor 3x \rfloor = 2n(3n+1) = 6n^2+2n$.
If $n+\dfrac{1}{2} \le x < n+\dfrac{2}{3}$ for some integer $n$, then $\lfloor 2x \rfloor \lfloor 3x \rfloor = (2n+1)(3n+1) = 6n^2+5n+1$.
If $n+\dfrac{2}{3} \le x < n+1$ for some integer $n$, then $\lfloor 2x \rfloor \lfloor 3x \rfloor = (2n+1)(3n+2) = 6n^2+7n+2$.
Now, you need to figure out how many integers between $1$ and $10000$ can be written in one of these forms.
