These are Homework question. They are pigeon hole principle questions and I have a very hard time with these unless I have worked on a similar problem before.
Q.1. Prove that if we select 87 numbers from the set $S = {1,2,3,....,171}$ then there are at least two consecutive numbers in our selection.
Q.2. Let $n\ \epsilon \ Z^+$. Show that there exists $a,b \ \epsilon \ Z^+$ with $a \neq b$ such that $n^a-n^b $is divisible by $10$
Any help, hints would be great.