Proof
(related to Proposition: Number of Multiples of a Given Number Less Than Another Number)
- Let $0 < n,k < m$ be natural numbers.
- We have $0 < k$ and $kn\le m$ if and only if $0 < k \le \frac mn.$
- By the definition of the floor function, the number of natural numbers $\le x$ is given by $\lfloor x \rfloor$ for any given real number $x > 0.$
- Therefore, if we set $k=\left\lfloor \frac mn\right\rfloor,$ we get the number of the positive multiples of $n$ smaller or equal $m.$
∎
Thank you to the contributors under CC BY-SA 4.0! 
- Github:
-
References
Bibliography
- Scheid Harald: "Zahlentheorie", Spektrum Akademischer Verlag, 2003, 3rd Edition
- Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927
