This is my first post.

I'm having a bit of trouble getting started with some questions in a text I am going though. Most of the questions are similar,so if I can get through one of them, I'm assuming it would serve as a guide to getting through the rest. Here are a few of the presented questions:

- Let a ∈ Zn. Show that a and n are relatively prime if and only if there exists an element b ∈ Zn such that ab=1.

I realize that to show that numbers are relatively prime, I have to show that gcd(a,n)=1 -- I guess using Euclidean Algorithm, which I know how to do with actual numbers.

- Let a ∈ Zn. Show that a and n are relatively prime if and only if a's row in the multiplication table of Zn contains every element of Zn.

- Show that n is a prime number if and only if in the multiplication table for Zn, except for the first row, all elements of Zn appear in each row.

I understand what the problem is asking, but I have no idea how to begin proving it.

If someone can shed some light or provide some guidance, it is greatly appreciated!