$f$ being irreducible over $\mathbb{Z}_{p}$ implies all equivalent polynomials of $f$ in $\mathbb{Z}[x]$ will be irreducible over $\mathbb{Z}$

Question

$f$ being irreducible over $\mathbb{Z}_{p}$ implies all equivalent polynomials of $f$ in $\mathbb{Z}[x]$ will be irreducible over $\mathbb{Z}$

I think $p$ is supposed to be a prime for the only following reason. I am explaining the reason by taking $p =10$.

If $p = 10$ then the statement would not be true for the element $10$..Because $10$ is a reducible element in $ \mathbb Z$ but so is not in $\mathbb{Z}_{p}$.

I think this is the only reason we suppose $p$ a prime number.

Am I correct? Is there any other reason? Can anyone please help me?