Let $d$ be a positive integer and consider any set $A$ of $d+1$ positive integers. Show that there exists two different numbers $x, y\ \epsilon\ A$ so that $ x \mod\ d = y \mod\ d$ and $x =/= y$.
Remainders after division by $d$ must fall in the range $0 ≤ r ≤ d − > 1$; there are precisely $d$ different remainders. By the pigeonhole principle, at least two of any collection of $d + 1$ integers must have the same remainder when divided by $d$.
I know I have to use the pigeon hole principle for this, but I'm unsure as to how to start this. Any help would be great