elementary number theory - Is this proof, that $sqrt{n}$ is irrational for all non-square $n in mathbb{N}$, correct or not?

Prove that the square root of all non-square numbers $n \in \mathbb{N}$ is irrational

I have made an attempt to prove this, I don't know if it's correct though:

Take a non-square number $n \in \mathbb{N}$, and we'll assume that $\sqrt{n}$ is rational.

$\sqrt{n} = \dfrac{p}{q}$ , $p,q \in \mathbb{N}$ and they have no common factors.

$$nq^2=p^2$$ Lets say that $z$ is a prime factor of $q$, it must also be a prime factor of $q^2$. However, it then must ALSO be a prime factor of $p^2$ because of the equality above, and this is a contradiction.