Let:
- $a,b,c,d$ be integers with $\frac{a}{b} - \frac{c}{d} > 0$
- $b \nmid a$ and $d \nmid c$
Does it follow that:
$$\left\lfloor\frac{a}{b}\right\rfloor - \left\lfloor\frac{c}{d}\right\rfloor = \left\lceil\frac{a}{b}\right\rceil - \left\lceil\frac{c}{d}\right\rceil$$
Here's my thinking:
There exists positive integers $e<b,f<d$ such that:
$$\left\lfloor\frac{a}{b}\right\rfloor - \left\lfloor\frac{c}{d}\right\rfloor = \frac{a-e}{b} - \frac{c-f}{d} = \frac{ad-ed - bc + bf}{bd}$$
$$\left\lceil\frac{a}{b}\right\rceil - \left\lceil\frac{c}{d}\right\rceil = \frac{a+(b-e)}{b} - \frac{c+(d-f)}{d} = \frac{ad+bd-ed - bc -bd + bf}{bd} = \frac{ad-ed - bc + bf}{bd}$$
Did I make a mistake? Am I wrong?
Your second condition says that the ratios $\dfrac ab$ and $\dfrac cd$ are not integers. Hence, $$\left\lceil\dfrac ab\right\rceil=\left\lfloor\dfrac ab\right\rfloor+1$$ $$\left\lceil\dfrac cd\right\rceil=\left\lfloor\dfrac cd\right\rfloor+1$$ Subtract the second equation from first and get the desired result.