A floor function problem

Any solution of the equation must be an integer. Any integer $x$ can be written uniquely as $x=30m+k$ for integers $m,k$ where $0\le k \le 29$ . The equation now becomes $31m + \big\lfloor \tfrac{k}{2}\big\rfloor + \big\lfloor\tfrac{k}{3}\big\rfloor +\big\lfloor\tfrac{k}{5}\big\rfloor = 30m + k$ so $m= k-\big\lfloor\tfrac{k}{2}\big\rfloor - \big\lfloor\tfrac{k}{3}\big\rfloor -\big\lfloor\tfrac{k}{5}\big\rfloor$ and we see that $m$ is uniquely determined by $k$ . Thus there is exactly one solution $x$ for each integer $0\le k\le 29$ , and so there are $\boxed{30}$ solutions.