Prime and rational numbers I came across the following question while studying that is stumping me. Can anyone please help me solve it? 
Let "$a$" be a prime number greater than $10,000$ and let $x=\sqrt{a}$. Which of the following expressions represents a rational number?
F) $x/2$
G) $\sqrt{x}$
H) $2x$
J) $x^2$
K) $x+2 $

Assume $x$ is rational. Then we can express $x$ as:
$$x=\frac{p}{q}$$
where $p,q$ are co-prime integers. $x=\sqrt a$ can be written as:
$$x^2=a$$
$$\frac{p^2}{q^2}=a$$
$$p^2=aq^2$$
As the prime $a$ divides the RHS, it divides the LHS too. $p=ka$ for some integer $k$. Substituting:
$$k^2a^2=aq^2$$
$$k^2a=q^2$$
Hence, $a$ also divides $q$, which implies $p$ and $q$ are not co-prime. Contradiction.
Thus, $x$ is irrational.
We can go ahead proving  by contradiction that F,G,H,K are all irrational.
J is the answer as $x^2=a$ is prime and thus integer and rational.   
