Alternate definition of prime number I know the definition of prime number when dealing with integers, but I can't understand why the following definition also works:

A prime is a quantity $p$ such that whenever $p$ is a factor of some product $a\cdot b$, then either $p$ is a factor of $a$ or $p$ is a factor of $b$.

For example, take $4$ (which clearly is not a prime): it is a factor of $16=8\cdot 2$, so I should check that either $4\mid 8$ or $4\mid 2$. But $4\mid 8$ is true. So $4$ is a prime, which is absurd.
Please note that English is not my first language, so I may have easily misunderstood the above definition.
Edit: Let me try formalize the definition as I understood it: $p$ is prime if and only if $\exists a\exists b(p\mid a\cdot b)\rightarrow p\mid a\lor p\mid b$.

Note that this is whenever.
$4|4 = 2\cdot 2$ however $4\nmid2$.
To the edited question:
No! The word "whenever" comes to say that $p$ is prime if $\forall a \forall b (p | a\cdot b \Rightarrow (p | a \vee p | b))$
