Number of Multiples less than Given Number

Theorem

Let $m, n \in \N_{\ge 1}$.

The number of multiples of $m$ not greater than $n$ is given by:

$q = \floor {\dfrac n m}$

where $\floor {\cdot}$ denotes the floor function

$(1): \quad n = q m + r$

where $0 \le r < q$.

As $r < q$, it follows that the greatest multiple of $m$ up to $n$ is $q m$.

So all the multiples of $m$ up to $n$ are:

$m, 2 m, 3 m, \ldots, q m$

Dividing both sides of $(1)$ by $m$:

$(2): \quad \dfrac n m = q + \dfrac r m$

Taking the floor of $(2)$:

$\floor {\dfrac n m} = \floor {q + \dfrac r m}$

But as $0 \le \dfrac r m < 1$:

$\floor {q + \dfrac r m} = q$

Recall that all the multiples of $m$ up to $n$ are $m, 2 m, 3 m, \ldots, q m$.

It follows that the number of multiples of $m$ up to $n$ is:

$q = \floor {\dfrac n m}$

$\blacksquare$