Suppose two integers $a,N$, where N is prime, is there a difference between requiring $gcd(a,N)=1$ and $N \not\mid \!\!\;a $?

Question

This is probably painfully obvious but I wanted to confirm if there's any difference between requiring that the $gcd(a,N)=1$ or $N \not\mid \!\!\;a $ if N is prime? That is, could you use either requirement and achieve the same result?

Answer 1

There is no difference, each requirement implies the other. If $\gcd(a,N)=1$ then obviously $N$ can't divide $a$ because otherwise $N$ would be a common divisor of $a,N$ which is bigger than $1$.

Now suppose $N$ does not divide $a$. Let $d$ be a positive common divisor of $a,N$. Since $N$ is prime we know $d=N$ or $d=1$. But $N$ does not divide $a$, hence $d=1$. So $1$ is the only positive common divisor of $a,N$, hence it is the $\gcd$.

Answer 2

Negate: $ $ prime $\, \color{#c00}{p\mid a} \iff p\mid a,p\iff \overbrace{p\mid (a,p)\iff \color{#c00}{(a,p)\neq 1}}^{\textstyle \hphantom{p\mid (a,p)} \ {\Longleftarrow\ \ (a,p)\mid p}}$