Proof of $x$ is divisible by $11$, if $x^2$ is divisible by $11$

Question

How to prove that $11\mid x$ if $11\mid x^2$ ? Is it enough to say that $11$ is just a prime number?

Answer 1

Yeah, it's enough. One of the basic properties of prime numbers is that $p|ab \implies p \mid a$ or $p \mid b$. For your example just let $a=b=x$

Answer 2

Another way to think of this is: is $11$ a square? Since $11$ divides a square number, namely $x^2$, then if it is a square itself, it's square root may, or may not divide $x$.

We see that since $11$ is not a square, and $11\mid x^2$, what this really tells us is that $11^2\mid x^2$ and so we must have $11\mid x$.