Prove that $n|(a-b)$, such that $a,b \in S$ and pick any $n + 1$ numbers from set $S$ using divisibility and Pigeonhole principle

Question

let $n$ be a positive interger , prove that if you pick any $n + 1$ numbers from any set $S$, there are two distinct numbers $(a,b) \in S$ (in the set) such that $n|(a-b)$.

So, I tried to prove that two of these $(a-b)$ have the same remainder , so their difference is divisible by n. I used divisibility and Pigeonhole principle but I m not really sure how to prove it.
Any suggestion is welcome.Thank you.

Answer 1

I am not sure what $S$ is. The integers$\mod n$ have $n$ distinct equivalence classes (represented by remainders). By the pigeonhole principle, if we take $n+1$ elements in $\mathbb{Z_{n}}$=$S?$, two of them must have the same remainder mod $n$. Then the difference of the two is a multiple of $n$.