Representation as difference

We can write $n=kd+1$ where $k$ is a positive integer.

According to the pigeonhole principle there is a least $\lceil \frac{n}{d} \rceil=k+1$

integers in $X$ witch have the same residue $r$ modulo $d$ .

Denote $dq_1+r<dq_2+r<\dots<dq_{k+1}+r$ those elements.

Let $\delta=\min \limits_{1\le i\le k}(q_{i+1}-q_i)$ .

We have:

$k\delta \le \sum\limits_{i=1}^{k}(q_{i+1}-q_i)=q_{k+1}-q_1=\frac{1}{d}((dq_{k+1}+r)-(dq_1+r))\le\frac{1}{d}(m-1)$ .

Wich means: $\delta \le \frac{1}{kd}(m-1)=\frac{m-1}{n-1}$ hence $\delta \le \lfloor \frac{m-1}{n-1} \rfloor=1$ but $\delta$ is a positive integer.

So $\delta=1\Rightarrow \exists l \in \{1,2,\dots,k\}; 1=q_{l+1}-q_l$ .

Now considere the numbers $a=dq_{l+1}+r$ and $b=dq_l+r$ .

We have clearly $a-b=d$ .