Let $[x]$ denote the greatest integer less than or equal to $x$ for $x \in \mathbb{R}$.
I started by exploring the following question:
Is $$ \left[\frac{x}{n}\right]-\left[\frac{y}{n}\right] = \left[\frac{x-y}{n}\right] $$
true where $x, y, n$ are positive integers and $x \ge n \ge y$?
I worked some small examples.
Case $x = 5, y = 3, n = 4$:
$$\left[\frac{x}{n}\right]-\left[\frac{y}{n}\right] = \left[\frac{5}{4}\right]-\left[\frac{3}{4}\right] = 1 - 0 = 1$$
$$\left[\frac{x-y}{n}\right] = \left[\frac{5-3}{4}\right] = \left[\frac{2}{4}\right] = 0$$
Case $x = 11, y = 4, n = 5$:
$$\left[\frac{x}{n}\right]-\left[\frac{y}{n}\right] = \left[\frac{11}{5}\right]-\left[\frac{4}{5}\right] = 2 - 0 = 2$$
$$\left[\frac{x-y}{n}\right] = \left[\frac{11-4}{5}\right] = \left[\frac{7}{5}\right] = 1$$
So, the statement is false under the given inequality condition $x \ge n \ge y$.
Question: Under what conditions is the statement
$$ \left[\frac{x}{n}\right]-\left[\frac{y}{n}\right] = \left[\frac{x-y}{n}\right] $$
true for $x,y,n$ positive integers, $n \gt 0$?
Let $X=x/n$ & $Y=y/n$ which are rationals.
$Z_1 = \left[\frac{x}{n}\right]-\left[\frac{y}{n}\right] = \left[X\right]-\left[Y\right]$
$Z_2 = \left[\frac{x-y}{n}\right] = \left[\frac{x}{n}-\frac{y}{n}\right] = \left[X-Y\right]$
Let $X=X_I+X_f$ where $X_I$ is Integer Part & $X_f$ is fractional Part.
Similarly , let $Y=Y_I+Y_f$ where $Y_I$ is Integer Part & $Y_f$ is fractional Part.
$Z_1=X_I-Y_I$
$Z_2=\left[X_I+X_f-Y_I-Y_f\right]$
If we want $Z_1=Z_2=X_I-Y_I$ , then $X_f-Y_f$ must be the fractional Part in $Z_2$.
Hence $X_f-Y_f$ is Zero or Positive. ( If it is Negative , $Z_2$ will reduce by $1$ )
Criteria 1 : Hence fractional Part of $X$ must be Equal to or larger than fractional Part of $Y$. [[ rational number Case ]]
Examples Satisfying Criteria 1 : We can check with $(X,Y)=(8.6,1.3)$ , $[8.6]-[1.3]=8-1=7$ & $[8.6-1.3]=[7.3]=7$
Examples Not Satisfying Criteria 1 : We can check with $(X,Y)=(8.6,1.7)$ , $[8.6]-[1.7]=8-1=7$ & $[8.6-1.7]=[6.9]=6$
We have to check the fractional Part , while the Integer Part can be arbitrary.
We can go back to Integers $x$ & $y$ :
$x \bmod n$ will contribute to the fractional Part.
$y \bmod n$ will contribute to the fractional Part.
Combined Contribution must be Zero or Positive ( when it is Negative , it will make $Z_2$ less by $1$ )
Criteria 2 : Hence $x \bmod n$ must be Equal to or larger than $y \bmod n$ [[ Integer Case ]]
Examples Satisfying Criteria 2 : We can check with $(x,n,y)=(86,8,3)$ , $[86/8]-[3/8]=10-0=10$ & $[(86-3)/8]=[83/8]=10$
When we do not want $y$ to be the smallest , we can just add multiples of $y=8$ to it & generate more Examples like $(x,n,y)=(86,8,19)$ & $(86,8,99)$ & $(86,8,899)$
Examples Not Satisfying Criteria 2 : We can check with $(x,n,y)=(86,8,7)$ , $[86/8]-[7/8]=10-0=10$ & $[(86-7)/8]=[79/8]=9$
When we do not want $y$ to be the smallest , we can just add multiples of $y=8$ to it & generate more Examples like $(x,n,y)=(86,8,15)$ & $(86,8,815)$ & $(86,8,831)$
Moving to the last requirement : We are given that $y$ is less than $n$ ( which is less than $x$ ) , hence $y \bmod n = y$
Combined value must not be Negative to ensure that $Z_2$ is matching $Z_1$.
Criteria 3 : Hence $x \bmod n$ must be Equal to or larger than $y$ [[ Integer Case with extra condition that $n$ is between $x$ & $y$ ]]
We have to take Examples for Criteria 2 & exclude the cases where $y$ is too large , nothing more to consider here.
Examples Satisfying Criteria 3 : We can check with $(x,n,y)=(86,8,3)$ , $[86/8]-[3/8]=10-0=10$ & $[(86-3)/8]=[83/8]=10$
Examples Not Satisfying Criteria 3 : We can check with $(x,n,y)=(86,8,7)$ , $[86/8]-[7/8]=10-0=10$ & $[(86-7)/8]=[79/8]=9$
That Criteria & Examples cover all Integer Cases & rational Cases.