How to find the values that make this interval enclose an integer

Question

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?

Answer 1

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 $n$.

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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 $n=\left\lceil\dfrac{q+1}{p-q}\right\rceil$.

Answer 2

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.