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.

*

*Let $n$ be a natural number.

*Suppose $n > 1$.

*n is prime if

*

*for all positive integers $p$

*where $p$ is prime

*and $p \leq \sqrt{n}$,

*$p$ does not divide $n$



*and only if

*

*for all positive integers $p$

*where $p$ is prime

*and $p \leq \sqrt{n}$,

*$p$ does not divide $n$
For instance, $17$ is prime if (and only if) all the primes in the interval $[1,\sqrt{17}]$ do not divide $17$.  The primes in the interval $[1, \sqrt{17}]$ are $2$ and $3$.  Since $2 \not\mid 17$ and $3 \not\mid 17$, $17$ is prime.
