Suppose a in an integer, and p is prime. Prove that if $gcd(a,p)>1$, then $p$ divides $a$

Question

Okay, since p is a prime and it is divisible by some number that is greater than 1, it has to be p itself. So if a and p have the same gcd, it has to be p, which implies that a has to be divisible by p. Is this enough proof or there is something else to say?

Answer 1

HINT: The positive divisors of $p$ are $1$ and $p$. As $1$ is ruled out we have that $(a,p) = p$.

And yes your proof seems to be alright.