I can't figure out how to prove the following:
Let $\displaystyle p \in N$ be a prime and $\displaystyle c \in Z$. Show that either $\displaystyle p | c$ or $\displaystyle gcd(p,c)=1$.
Any help is appreciated.
You need lessons in basic logic.
The point is that either $\displaystyle k=\gcd(p,c)=1 \text{ or } k=\gcd(p,c)>1$.
Now if $\displaystyle k=\gcd(p,c)=1$ we are done with the proof.
So suppose that $\displaystyle k=\gcd(p,c)>1$ then $\displaystyle k \not= 1$.
Thus this means that $\displaystyle k=p$ so $\displaystyle p|c$.