"How many numbers between $1$ and $n$ — inclusive — are multiples of $m$?". Perhaps it's very easy to find the answer to such a basic question; hence I do not need an answer to the question, but rather an explanation. Let us conjure an example:
"How many numbers between $1$ and $200$ — inclusive — are divisible by $6$?". According to formulas derived by Gauss, if we consider the sequence $6, ..., 114$ (the first term taken to be the smallest number between $1$ and $200$ that is divisible by $6$, and the last term taken to be the largest number between $1$ and $200$ that is divisible by $6$), then the answer to the question proposed is identical to the number of terms in the sequence we just crafted. The answer is easily drawed from the equation $6 + 6(n – 1) = 114$, hence $n$ (the number of terms) $= 19$.
However, it is foolish to over this routine everytime we try and resolve such a common question. Thankfully, in Brauldi's book on Introductory Combinatorics, he shows that the number of positive integers between $1$ and $n$ — inclusive — that are divisible by $m$ is $ [$$n\over m$$]$, where $[x]$ is the floor (greatest integer in) function.
I would love to start using this quicker method in resolving such questions, however, I struggled to intuitively figure out why exactly it always works. It does make sense, but I still don't have a good idea on why it's true. I request you, reader, to assist my thought process with elucidation so I can understand the picture more lucidly. Thank you in advance.