🤣 Floor Division Reveals Number of Divisible Numbers

Take something like $\lfloor \dfrac{13}{4} \rfloor$ for example

$$ 1, \ 2, \ 3, \ 4, \ 5, \ 6,\ 7,\ 8,\ 9,\ 10,\ 11,\ 12, \ 13 $$

This will be grouped into boxes of size $4$

$$ \boxed{1, \ 2, \ 3, \ 4,} \ \boxed{5, \ 6,\ 7,\ 8,} \ \boxed{9,\ 10,\ 11,\ 12,} \ \boxed{13 \qquad \qquad \phantom{,} } $$

Floor division will remove the groups that aren’t full

$$ \boxed{1, \ 2, \ 3, \ 4,} \ \boxed{5, \ 6,\ 7,\ 8,} \ \boxed{9,\ 10,\ 11,\ 12,} \ \phantom{\boxed{13 \qquad \qquad \phantom{,} }} $$

Then each multiple of $4$, will be the representing number for each group

$$ \boxed{1, \ 2, \ 3, \ \textcolor{darkgreen}{4},} \ \boxed{5, \ 6,\ 7,\ \textcolor{darkgreen}{8},} \ \boxed{9,\ 10,\ 11,\ \textcolor{darkgreen}{12},} \ \phantom{\boxed{13 \qquad \qquad \phantom{,} }} $$

Thus, the number of groups is also the number of numbers divisible by $4$

---

This is a rule true for finding the number of numbers divisible by any number $m$, up to number $n$

$$ \text{Number of numbers divisible by m} = \lfloor \dfrac{n}{m} \rfloor $$

---

And if we want to find the numbers between $n_{1}$ and $n_{2}$ divisible by $m$, given $n_2$ > $n_1$, we simply find the difference of the floor divisions

$$ \text{Number of numbers divisible by m between} \ n_{2} \ \text{and} \ n_{1} = \lfloor \dfrac{n_2}{m} \rfloor - \lfloor \dfrac{n_1}{m} \rfloor $$