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

Proof
By the Division Theorem:


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

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

Also see

 * De Polignac's Formula