Pigeonhole Principle Problem - need help understanding the concept

Question

Let $n ∈ \mathbb{N}$ be a natural number and let $X ⊆ \mathbb{N}$ be a subset with $n + 1$ elements. Show that there exist two natural numbers $x, y ∈ X$ such that $x − y$ is divisible by $n$.

Answer 1

If you have heard of modular arithmetic that will help here. The idea is to put each element of $X$ into a class labeled $0$ to $n-1$, where $x$ goes in class $m$ if the remainder of $x$ divided by $n$ is $m$. If two numbers $x$ and $y$ are in the same class, they will differ by a multiple of $n$, and so $x-y$ will be divisible by $n$. By the pigeonhole principle, you have $n+1$ elements and $n$ classes, so two elements must be in the same class, and thus they satisfy $x-y$ is divisible by $n$.

Answer 2

We prepare $n$ boxes which means dividing by $n$. If there are $n+1$ elements to put into the $n$ boxes, there must be at least one box with at least 2 elements.