If $p$ is a prime number and $a \in \mathbb{Z}$ isn't multiple of $p$, can we then state that the gcd(a,p)=1?

Question

So suppose that $p$ is a prime number, and that $\forall n \in \mathbb{Z}, a \not = np$, then can we state that the gcd (greatest common divisor) is equal to $1$?

Answer 1

Yes. The only divisors of $p$ are $1$ and $p$ (because it's prime), and $\gcd(a,p)$ must divide $p$ so has to be either $1$ or $p$. But also $\gcd(a,p)$ must divide $a$. Since $p$ doesn't divide $a$, the only possibility remaining is $\gcd(a,p)=1$.