How to find the values that make this interval enclose an integer Suppose I have an interval that looks like this:
$\left[\frac{k}{\lfloor m \rfloor }, \frac{k}{\lfloor mr \rfloor}\right)$
$m$ and $r$ are positive real numbers, but are constants in this problem.
Here is the question:
Which integer values of $k$ make this interval include at least one integer?
I have tried various ways to tackle this problem, but none have panned out.  The thing is I have no idea where to even start.  How does one solve problems like this?
 A: Let $n$ be the enclosed integer. Assuming $p=\lfloor m\rfloor>q=\lfloor mr\rfloor$, we have
$$\frac kp\le n<\frac kq$$ so that
$$nq<k\le np$$ for all positive $n$.

These intervals go growing with $n$ and start touching when
$$np>(n+1)q$$ or 
$$n(p-q)>q.$$
When this condition is met, every $k>nq$ is a good fit.
Hence
$$k\in(q,p]\cup(2q,2p]\cup(3q,3p]\cup\cdots(nq,\infty)$$ where $n=\left\lceil\dfrac{q+1}{p-q}\right\rceil$.
A: You can think of this graphically. Consider the graphs of the functions
$f(x) = \frac{1}{\lfloor m\rfloor} x$
and $g(x) = \frac{1}{\lfloor mr\rfloor} x$
in a Cartesian plane with horizontal coordinate $x.$
Now consider all of the points with integer coordinates that lie on or above
the graph of $f(x)$ but below the graph of $g(x).$
The set of $k$ you are looking for is the set of $x$ coordinates of all those points.
Clearly, for $k$ large enough that the difference $g(k) - f(k)$ is at least $1,$ the interval 
$\left[\frac{k}{\lfloor m \rfloor }, \frac{k}{\lfloor mr \rfloor}\right)$
contains at least one integer.
You just have to check the  finite number of values of $k$ for which 
$g(k) - f(k) < 1$ in order to find all remaining values of $k$ for which
$\left[\frac{k}{\lfloor m \rfloor }, \frac{k}{\lfloor mr \rfloor}\right)$
contains at least one integer.
