Prove identity composed of floor functions

Question

How to prove: $$\left\lfloor \frac{x+2}{6}\right\rfloor -\left\lfloor \frac{x+3}{6}\right\rfloor +\left\lfloor \frac{x+4}{6}\right\rfloor =\left\lfloor \frac{x}{2}\right\rfloor -\left\lfloor \frac{x}{3}\right\rfloor$$

where $x\in \mathbb{R}$. I think it is problem made by Ramanujan, but do not have the source.

Answer 1

Hint: write $x= 6k+r$ where $o\leq r <6$ and $k\in \mathbb{Z}$. Then we have:

$$\left\lfloor \frac{x+2}{6}\right\rfloor -\left\lfloor \frac{x+3}{6}\right\rfloor +\left\lfloor \frac{x+4}{6}\right\rfloor = k + \underbrace{\left\lfloor \frac{r+2}{6}\right\rfloor -\left\lfloor \frac{r+3}{6}\right\rfloor +\left\lfloor \frac{r+4}{6}\right\rfloor}_{A}$$

and

$$\left\lfloor \frac{x}{2}\right\rfloor -\left\lfloor \frac{x}{3}\right\rfloor = k +\underbrace{\left\lfloor \frac{r}{2}\right\rfloor -\left\lfloor \frac{r}{3}\right\rfloor}_{B}$$