(floor function) sum of x: $\left\lfloor{\frac{x}{5}}\right\rfloor - \left\lfloor{\frac{x}{9}}\right\rfloor = \frac{x}{15}$
Solution 1:
First, since the LHS is an integer, we know $x$ is a multiple of $15$. Thus, let $x = 15n$. Then our equation reduces to $$n = 3n - \left\lfloor \frac{15n}{9}\right\rfloor \implies$$ $$2n = \left\lfloor \frac{15n}{9}\right\rfloor \leq \frac{15n}{9}$$ thus ruling out any positive solutions. Now note that $$2n = \left\lfloor \frac{15n}{9}\right\rfloor \geq \frac{15n}{9}-1 \implies$$ $$n \geq -3$$ So any possible solutions for $n$ satisfy $-3 \leq n \leq 0$. Checking manually, we see that the only solutions are $n = 0, -1, -2$, i.e. $x = 0, -15, -30$.