Understanding basic concept of prime numbers My textbook provides a theorem but I cannot understand the structure of the sentence being used. Could someone please help me understand the meaning of this theorem?

A natural number $n>1$ is prime if and only if for all primes $p\leq \sqrt{n}$, $p$ does not divide $n$.

"for all primes $p\leq \sqrt{n}$, $p$ does not divide $n$". This is the part I don't understand. The premise is that $p$ is less than the square root of $n$ then the conclusion is that $p$ does not divide $n$. So for $p$ to divide $n$ it must be greater than the square root of $n$? I'm really confused as to what information and meaning I'm supposed to see in this theorem.

It says that, in order to see if $n$ is prime, we don't have to test divide $n$ by every prime less than $n$.  It's enough to check all primes $\leq\sqrt n$.  For example, if we want to know if $101$ is prime, it's enough to check that it's not divisible by $2,3,5,\text{ or }7$, because the next prime, $11$ is $>\sqrt{101}$.
This is because if $n=ab$, it's impossible that both $a>\sqrt n$ and $b>\sqrt n$.
