<s>[INST] Multiplying a floor function to a number Is it correct to write: 
$\cfrac{\left\lfloor{\cfrac{\pi y^2}{3\sqrt{3}x^2}}\right\rfloor}{n} \times\sqrt{3}x =\left\lfloor\cfrac{\pi y^2}{3xn}\right\rfloor$ ?
 [/INST] No, it is not correct.
For $x,y$ such that $\left\lfloor\frac{\pi y^2}{3\sqrt 3 x^2}\right\rfloor\not=0$, the LHS is irrational and the RHS is rational.
Added : 

when could this be true?

We need $$\left\lfloor\frac{\pi y^2}{3\sqrt 3 x^2}\right\rfloor=\left\lfloor\frac{\pi y^2}{3xn}\right\rfloor=0,$$
i.e.
$$0\le \frac{\pi y^2}{3\sqrt 3 x^2}\lt 1\ \ \ \text{and}\ \ \ 0\le \frac{\pi y^2}{3xn}\lt 1,$$
i.e.
$$\pi y^2\lt 3\sqrt 3x^2\ \ \ \text{and}\ \ \ \pi y^2\lt 3xn.$$
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