Hey, I am having a difficult time trying to prove a few statements relating to Euler's Theorem.

1. Prove for any n: If $\displaystyle x,y$ are in $\displaystyle \Phi_n$ , then $\displaystyle xy \ mod \ n$ is also in $\displaystyle \Phi_n$, using a contrapositive argument.

2. Prove: Every $\displaystyle a \in \Phi_n$ has an inverse modulo n in $\displaystyle \Phi_n$ , by considering the linear congruence $\displaystyle ax \equiv 1 (mod \ n)$

3. Prove: For every $\displaystyle a \in \Phi_n$ the function $\displaystyle f_a : \Phi_n \to\Phi_n$ defined by $\displaystyle f_a(x) =ax$ is a permutation.

I really don't know what to do for these.

Any help would be greatly appreciated.

Thanks